análisis-matemático y medicina y ley

Teorema:

Sea a_{n} acotada ==>

Si [An][ |b_{n}| [< |a_{n}+c| ] ==> b_{n} está acotada

Demostración:

Sea a_{n} acotada ==>

[EN][An][ |a_{n}| [< N ]

|b_{n}| [< |a_{n}+c| [< |a_{n}|+|c| [< N+|c|

Se define M = N+|c| ==>

|b_{n}| [< M

Teorema:

[An][ 0 [< e^{n} & 0 [< e^{(-n)} ]

Demostración:

por inducción:

0 [< 1 = e^{0}

0 [< 1 [< e^{n} [< e^{n+1}

Por descenso:

0 = (1/e^{oo}) = ( 1/oo^{[e]+(-1)} ) = (1/oo)

0 [< (1/e^{n}) [< (1/e^{n+(-1)})

Teorema:

Sea a_{n} acotada ==>

Si [An][ |b_{n}| [< |e^{|a_{n}|}+c| ] ==> b_{n} está acotada

Demostración:

Sea a_{n} acotada ==>

[EN][An][ |a_{n}| [< N ]

e^{|a_{n}|} [< e^{N}

|b_{n}| [< |e^{|a_{n}|}+c| [< |e^{|a_{n}|}|+|c| = e^{|a_{n}|}+|c| [< e^{N}+|c|

Se define M = e^{N}+|c| ==>

|b_{n}| [< M

Teorema:

[As][ Si s >] 0 ==> ln(s+1) >] 0 ]

Demostración:

Sea s >] 0 ==>

s+1 >] 1

ln(s+1) >] ln(1) = 0

Teorema:

Sea a_{n} acotada ==>

Si [An][ |b_{n}| [< |ln( |a_{n}|+1 )+c| ] ==> b_{n} está acotada

Demostración:

Sea a_{n} acotada ==>

[EN][An][ |a_{n}| [< N ]

|a_{n}|+1 [< N+1

ln( |a_{n}|+1 ) [< ln(N+1)

|b_{n}| [< |ln( |a_{n}|+1 )+c| [< |ln( |a_{n}|+1 )|+|c| = ln( |a_{n}|+1 )+|c| [< ln(N+1)+|c|

Se define M = ln(N+1)+|c| ==>

|b_{n}| [< M

Teorema:

F(x,t) = ( x(t) )^{p+1}+(-1)·h(t)·( x(t)+(-1)·M(t) )

h(t) = (p+1)·( M(t) )^{p}

Demostración:

d_{x}[ F(x,t) ] = d_{x}[ ( x(t) )^{p+1}+(-1)·h(t)·( x(t)+(-1)·M(t) ) ] = ...

... d_{x}[ ( x(t) )^{p+1} ]+d_{x}[ (-1)·h(t)·( x(t)+(-1)·M(t) ) ] = ...

... d_{x}[ ( x(t) )^{p+1} ]+(-1)·h(t)·d_{x}[ x(t)+(-1)·M(t) ] = ...

... d_{x}[ ( x(t) )^{p+1} ]+(-1)·h(t)·( d_{x}[ x(t) ]+d_{x}[ (-1)·M(t) ] ) = ...

... (p+1)·( x(t) )^{p}+(-1)·h(t) = 0

Teorema:

F(x,t) = e^{(p+1)·x(t)}+(-1)·h(t)·( e^{x(t)}+(-1)·M(t) )

h(t) = (p+1)·( M(t) )^{p}

Teorema:

F(x,y,t) = ( x(t) )^{p+1}+( y(t) )^{p+1}+(-1)·h(t)·( ( x(t)+y(t) )+(-1)·M(t) )

h(t) = (1/2)·(p+1)·( ( (1+(-1)·k(t))·M(t) )^{p}+( k(t)·M(t) )^{p} )

Teorema:

F(x,y,t) = e^{(p+1)·x(t)}+e^{(p+1)·y(t)}+(-1)·h(t)·( ( e^{x(t)}+e^{y(t)} )+(-1)·M(t) )

h(t) = (1/2)·(p+1)·( ( (1+(-1)·k(t))·M(t) )^{p}+( k(t)·M(t) )^{p} )

Teorema:

F(x,y,t) = x+y+(-1)·h(t)·( (x+y)+(-1)·M(t) )

h(t) = 1

Teorema:

F(x,y,t) = e^{x}+e^{y}+(-1)·h(t)·( (e^{x}+e^{y})+(-1)·M(t) )

h(t) = 1

Teorema:

F(x,y,t) = ( x^{2}+y^{2} )+(-1)·h(t)·( (x+y)+(-1)·M(t) )

h(t) = M(t)

Demostración:

2x+2y = 2·M(t)

Teorema:

F(x,y,t) = ( e^{2x}+e^{2y} )+(-1)·h(t)·( (e^{x}+e^{y})+(-1)·M(t) )

h(t) = M(t)

Teorema:

F(x,y,t) = ( x^{2}+nxy+y^{2} )+(-1)·h(t)·( (x+y)+(-1)·M(t) )

h(t) = (1/2)·(n+2)·M(t)

Demostración:

2x+2y+n·(y+x) = (n+2)·M(t)

Teorema:

F(x,y,t) = ( e^{2x}+ne^{x+y}+e^{2y} )+(-1)·h(t)·( (e^{x}+e^{y})+(-1)·M(t) )

h(t) = (1/2)·(n+2)·M(t)

Ley:

Rezar al Mal matar no tiene sentido,

porque se hace Esparta matando que es vida. 

Rezar al Mal follar con una esclava infiel no tiene sentido,

porque se hace Esparta matando que es vida.

Ley:

Escudos electro-magnéticos en n = 1 en el Gauge blanco ==>

F(t)·G(t) = ( f(t)·g(t) )^{(1/2)·(n+(-1))+2}

Camuflaje gravito-magnético en n = 2 en el Gauge negro ==>

F(t)·G(t) = ( f(t)·g(t) )^{n}

Principio: [ de oncología ]

Sea ( K(t) el cabal del fluido intestinal & d_{xyz}^{3}[m(x,y,z)] su densidad ) ==>

[EM][ M(x,y,z,t) = int[ d_{xyz}^{3}[m(x,y,z)]·K(t) ]d[t] ]

Defecación Normal t >] (1/u):

Ley:

Sea ( K(t) = Vu·(ut)^{n} & d_{xyz}^{3}[m(x,y,z)] = ma^{3} ) ==>

M(x,y,z,t) = ma^{3}·V·(1/(n+1))·(ut)^{n+1}

m(x,y,z) = ma^{3}·xyz

M(x,y,z,(1/u)) = ma^{3}·V·(1/(n+1))

Ley:

Sea ( K(t) = Vu·(1/(ut)) & d_{xyz}^{3}[m(x,y,z)] = ma^{3} ) ==>

M(x,y,z,t) = ma^{3}·V·ln(ut)

m(x,y,z) = ma^{3}·xyz

M(x,y,z,(1/u)) = 0

Defecación Encima t >] (0/u):

Ley:

Sea ( K(t) = Vu·(1+(ut))^{n} & d_{xyz}^{3}[m(x,y,z)] = ma^{3} ) ==>

M(x,y,z,t) = ma^{3}·V·(1/(n+1))·(1+(ut))^{n+1}

m(x,y,z) = ma^{3}·xyz

M(x,y,z,(0/u)) = ma^{3}·V·(1/(n+1))

Ley:

Sea ( K(t) = Vu·( 1/(1+(ut)) ) & d_{xyz}^{3}[m(x,y,z)] = ma^{3} ) ==>

M(x,y,z,t) = ma^{3}·V·ln(1+ut)

m(x,y,z) = ma^{3}·xyz

M(x,y,z,(0/u)) = 0

Defecación Normal con Retortijón ( vt >] (1/a) & t >] (1/u) ):

Ley:

Sea ( K(t) = Vu·(ut)^{n+1} & d_{xyz}^{3}[m(x,y,z)] = ma^{2}·(1/z) ) ==>

Si z = vt ==> M(x,y,z,t) = ma^{2}·V·(u/v)·(1/(n+1))·(ut)^{n+1}

m(x,y,z) = ma^{2}·xy·ln(az)

M(x,y,z,(1/u)) = ma^{2}·V·(u/v)·(1/(n+1))

Ley:

Sea ( K(t) = Vu & d_{xyz}^{3}[m(x,y,z)] = ma^{2}·(1/z) ) ==>

Si z = vt ==> M(x,y,z,t) = ma^{2}·V·(u/v)·ln(ut)

m(x,y,z) = ma^{2}·xy·ln(az)

M(x,y,z,(1/u)) = 0

Defecación Encima con Retortijón ( vt >] (1/a) & t >] (0/u) ):

Ley:

Sea ( K(t) = Vu·(1+(ut))^{n} & d_{xyz}^{3}[m(x,y,z)] = ma^{2}·(1/z) ) ==>

Si z = vt ==> M(x,y,z,t) = ma^{2}·V·(u/v)·( ln(ut) [o(ut)o] (1/(n+1))·(1+(ut))^{n+1} )

m(x,y,z) = ma^{2}·xy·ln(az)

M(x,y,z,(0/u)) = ma^{2}·V·(u/v)·(1/(n+1))·ln(2)

Ley:

Sea ( K(t) = Vu·( 1/(1+(ut)) ) & d_{xyz}^{3}[m(x,y,z)] = ma^{2}·(1/z) ) ==>

Si z = vt ==> M(x,y,z,t) = ma^{2}·V·(u/v)·( ln(ut) [o(ut)o] ln(1+(ut)) )

m(x,y,z) = ma^{2}·xy·ln(az)

M(x,y,z,(0/u)) = 0

Voy a estar yo como dios del Mal,

porque hay Júpiter como dios de los hombres,

y estos de ahora se van a extinguir saltando-se la Ley del Mal:

Hobbes.

Falso testimonio dual a la verdad.

No der ni dar energía nunca.

No hacer Esparta nunca.

Se saltan todas la Leyes del Mal:

Dicen en la mente que la gente no era,

me cago encima sabiendo oncología,

les den o les dan energía porque siguen hablando en la mente,

y tengo terror-ansiedad haciendo Esparta.

Teorema:

Si [Es(x)][ F( P(x)+Q(x) ) = P(x)+Q(x)+s(x) ] ==> F es lineal

Demostración:

[Es(x)][ F( P(x)+Q(x) ) = P(x)+Q(x)+s(x) = ( P(x)+(1/2)·s(x) )+( Q(x)+(1/2)·s(x) ) ]

Se define u(x) = v(x) = (1/2)·s(x) 

[Eu(x)][Ev(x)][ F( P(x)+Q(x) ) = ( P(x)+u(x) )+( Q(x)+v(x) ) = F( P(x) )+F( Q(x) ) ]

Teorema:

Si [Es(x)][ F( a·P(x) ) = a·P(x)+s(x) ] ==> F es lineal

Demostración:

[Es(x)][ F( a·P(x) ) = a·P(x)+s(x) = a·( P(x)+(1/a)·s(x) ) ]

Se define w(x) = (1/a)·s(x) 

[Ew(x)][ F( a·P(x) ) = a·( P(x)+w(x) ) = a·F( P(x) ) ]

Ley: [ de alcohólicos anónimos ]

Sea F( P(x) ) = P(x)+ax+(-b) el estado de drogadicción ==> ...

... F( P( (n+1)·(b/a) ) ) = P( (n+1)·(b/a) )+nb

Ker(F) = { Q(x) = (-1)·ax+b }

Q( (n+1)·(b/a) ) = (-1)·nb

Anexo:

La terapia solo funciona,

con n oyentes drogadictos según el Ker(F),

y se tiene que superar el tiempo,

de drogar-se acompañado a drogar-se solo,

con x = (b/a) para que el Ker(F) = { Q(x) = 0 },

y no haya estado de drogadicción,

con solo un (-b) de explicar la propia experiencia.

Ley: [ de tabaco con marihuana o de tabaco con hachís ]

Sea F( P(x) ) = P(x)+acx^{2}+(-b) el estado de drogadicción ==> ...

... F( P( ( (n+1)·(b/(ac) )^{(1/2)} ) ) = P( ( (n+1)·(b/(ac) )^{(1/2)} )+nb

Ker(F) = { Q(x) = (-1)·acx^{2}+b }

Q( ( (n+1)·(b/(ac) )^{(1/2)} ) = (-1)·nb

Anexo:

Con la marihuana y el hachís,

le decía a mi tío Carlos que me destruía,

y hacía terapia de fumar con mi padre,

hasta que lo he dejado.

Ley: [ de no tomar café cada día ]

Sea F( P(x) ) = P(x)+ax+(-b) el estado de drogadicción del café ==> ...

... F( P( ((2b)/a) ) ) = P( ((2b)/a) )+b

Ker(F) = { Q(x) = (-1)·ax+b }

Q( ((2b)/a) ) = (-b)

Anexo:

El (-b) del Ker(F) es igual a:

que la enfermera,

habiendo sido adicta al café,

me dijo que el Red-Bull era como 7 cafés,

y se mantiene la no adicción al café,

tomando café con la enfermera,

en ser solo 2 personas en el bar.

Ley: [ de droga única ]

En la calle no se puede fumar tabaco,

ni beber alcohol,

fuera de un parque con bancos o de una terraza de bar,

según sanidad,

porque el número de oyentes de drogadictos anónimos,

es descomunal.

Ley: [ de dos drogas ]

En la calle no se puede fumar porros,

ni beber alcohol con bebida energética,

fuera de un parque con bancos o de una terraza de bar,

según sanidad,

porque el número de oyentes de drogadictos anónimos,

es descomunal.

Anexo:

Fumar en una manifestación,

es de loco y está prohibido,

de tanta gente que hay,

y son años de fumar solo. 

Ley:

El intervalo legal de entrada en una discoteca con drogadicción,

es de 18 a 30 años.

El intervalo de prohibición de drogadicción en un establecimiento,

es de 30 a 42 años.

Anexo:

Te tienes que drogar solo durante 12 años,

después de drogar-te acompañado en la discoteca.

ley-y-filosofía y medicina y stowed-nipon-chinese y análisis-matemático y especies combinatorias y series-de-Fourier

Monarquía parlamentaria:

Artículo:

El poder legislativo ejecutivo,

reside en el Congreso de los Diputados,

y es escogido por sufragio universal.

El poder legislativo judicial,

reside en el Senado,

y es escogido por sufragio universal.

Artículo:

El presidente del poder ejecutivo,

es escogido por el Congreso de los Diputados.

El presidente del poder judicial,

es escogido por el Senado.

Artículo:

El Rey es árbitro de la democracia ejecutiva,

pudiendo inhabilitar diputados,

en el Congreso de los Diputados.

El Rey es árbitro de la democracia judicial,

pudiendo inhabilitar senadores,

en el Senado.

Republica Presidencialista:

Artículo:

El poder legislativo ejecutivo,

reside en el Congreso de los Diputados,

y es escogido por sufragio universal.

El poder legislativo judicial,

reside en el Senado,

y es escogido por sufragio universal.

Artículo:

El presidente del poder ejecutivo,

es escogido por sufragio universal.

El presidente del poder judicial,

es escogido por Sufragio universal.

Ley:

Sea ( x no creer-se Jesucristo & y creer-se Jesucristo ) ==>

Si ( a = 23 & b = 7 ) ==>

[05][...][05][...][05][03][05][...] = 23

[11][...][11][...][11][09][11][...] = 53

Falsus Algebratorum:

53 = 30+23 = 30+(-23) = 7

Fórmula:

=Br=(CH)-(Br|-|N)-(CH)=

Ley:

Sea ( x no se cree tu dios & y se cree tu dios ) ==>

Si ( a = 1 & b = 29 ) ==>

[05][...][05][...][05][09][05][...] = 29

[11][...][11][...][11][15][11][...] = 59

Falsus Algebratorum:

59 = 30+29 = 30+(-29) = 1

Fórmula:

=Br=(CH)-(Br|-|Krg|-|N)-(CH)=

Al primer año de voces no vas al psiquiatra,

porque la voz no se cree Jesucristo,

y después se lo cree durante 29 años.

Ley:

Sea ( x estar cerrado en el hospital & y no estar cerrado en el hospital ) ==>

Si ( a = 7 & b = 5 ) ==>

[06][...][01][...] = 07

[12][...][07][...] = 19

Falsus Algebratorum:

19 = 12+7 = 12+(-7) = 5

Fórmula:

=C=C=C-O-O-N=

Se convierte cerrar a un hombre,

en enfermedad mental de doble mandamiento,

en ser una distancia a la menos uno.

Stowed-Nipon-Chinese:

Dual:

Kino-yute [o] Kino-yang [o] Ayer

Hata-yute [o] Hata-yang [o] Mañana

Dual:

Sasa-yute [o] Sasa-yang [o] Matina

Yoro-yute [o] Yoro-yang [o] Tarde

Dual:

Hiro-yute [o] Hiro-yang [o] Medio-día

Banga-yute [o] Banga-yang [o] Noche

Dual:

Kioro-yute [o] Kioro-yang [o] Hoy

Hisa-yute [o] Hisa-yang [o] Día

Dual:

I stare-kate-maruto,

drinket-yuto-yaming mutchet-muto.

kioro-yute by sasa-yute.

I stareti-kate-maruto,

drinket-yuto-yaming pocket-muto,

kioro-yute by yoro-yute.

Dual:

I stare-kate-tai-tai,

drinket-yung-yanguing mutchet-tai-mung,

kioro-yang by sasa-yang.

I stareti-kate-tai-tai,

drinket-yung-yanguing pocket-tai-mung,

kioro-yang by yoro-yang.

Teorema:

Si f_{n}(x) es creciente ==> int[ f_{n}(x) ]d[x] es monótona

Si f_{n}(x) es decreciente ==> int[ f_{n}(x) ]d[x] es monótona

Demostración:

[1] Sea f_{n}(x) [< f_{n+1}(x) ==>

f_{n}(x)·d[x] [< f_{n+1}(x)·d[x]

int[ f_{n}(x) ]d[x] [< int[ f_{n+1}(x) ]d[x]

[2] Sea f_{n}(x) [< f_{n+1}(x) ==>

f_{n}(x)·d[x] >] f_{n+1}(x)·d[x]

int[ f_{n}(x) ]d[x] >] int[ f_{n+1}(x) ]d[x]  

Teorema: [ de convergencia monótona ]

Si f_{n}(x) es creciente ==> lim[n = oo][ int[ f_{n}(x) ]d[x] ] = int[ f(x) ]d[x]

Si f_{n}(x) es decreciente ==> lim[n = oo][ int[ f_{n}(x) ]d[x] ] = int[ f(x) ]d[x]

Demostración:

Sea f_{n}(x) es creciente ==>

int[ f_{n}(x) ]d[x] es monótona

int[ f_{n}(x) ]d[x] es creciente || int[ f_{n}(x) ]d[x] es decreciente  

[1] Si int[ f_{n}(x) ]d[x] es creciente ==>

int[ f_{n}(x) ]d[x] [< int[ f_{n+1}(x) ]d[x] [< int[ f(x) ]d[x]

lim[n = oo][ int[ f_{n}(x) ]d[x] ] [< int[ f(x) ]d[x]

Se define ( m = 1 || m = oo ) ==>

Sea n >] m ==>

int[ f_{n}(x) ]d[x] >] ( 1+(-1)·(1/m) )^{p}·int[ f(x) ]d[x]

lim[n = oo][ int[ f_{n}(x) ]d[x] ] >] int[ f(x) ]d[x]

[2] Si int[ f_{n}(x) ]d[x] es decreciente ==>

int[ f_{n}(x) ]d[x] >] int[ f_{n+1}(x) ]d[x] >] int[ f(x) ]d[x]

lim[n = oo][ int[ f_{n}(x) ]d[x] ] >] int[ f(x) ]d[x]

Se define ( m = 1 || m = oo ) ==>

Sea n >] m ==>

int[ f_{n}(x) ]d[x] [< ( 1+(-1)·(1/m) )^{p}·int[ f(x) ]d[x]

lim[n = oo][ int[ f_{n}(x) ]d[x] ] [< int[ f(x) ]d[x]

Teorema: [ de convergencia dominada ]

Si [Ek][An][ n >] k ==> f_{n}(x) >] f(x) ] ==> lim[n = oo][ int[ f_{n}(x) ]d[x] ] = int[ f(x) ]d[x]

Si [Ek][An][ n >] k ==> f_{n}(x) [< f(x) ] ==> lim[n = oo][ int[ f_{n}(x) ]d[x] ] = int[ f(x) ]d[x]

Demostración:

Sea n >] k ==> 

f_{n}(x) >] f(x)

f_{n}(x)·d[x] >] f(x)·d[x]

int[ f_{n}(x) ]d[x] >] int[ f(x) ]d[x]

Se define ( m = 1 || m = oo ) ==>

Sea n >] m ==>

int[ f_{n}(x) ]d[x] [< (1+(-1)·(1/m))^{p}·int[ f(x) ]d[x]

lim[n = oo][ int[ f_{n}(x) ]d[x] ] = int[ f(x) ]d[x]

Sea n >] k ==> 

f_{n}(x) >] f(x)

f_{n}(x)·d[x] [< f(x)·d[x]

int[ f_{n}(x) ]d[x] [< int[ f(x) ]d[x]

Se define ( m = 1 || m = oo ) ==>

Sea n >] m ==>

int[ f_{n}(x) ]d[x] >] (1+(-1)·(1/m))^{p}·int[ f(x) ]d[x]

lim[n = oo][ int[ f_{n}(x) ]d[x] ] = int[ f(x) ]d[x]

Teorema: [ de la primera convergencia algebraica ]

Si [Eh(x)][An][ f_{n}(x)+g_{n}(x) = h(x) ] ==> ...

... lim[n = oo][ int[ f_{n}(x) ]d[x] ] = int[ f(x) ]d[x] ...

... <==> ...

... lim[n = oo][ int[ g_{n}(x) ]d[x] ] = int[ g(x) ]d[x]

Teorema: [ de la segunda convergencia algebraica ]

Si [Eh(x)][An][ f_{n}(x)·g_{n}(x) = h(x) ] ==> ... 

... lim[n = oo][ int[ f_{n}(x) ]d[x] ] = int[ f(x) ]d[x] ...

... <==> ...

... lim[n = oo][ int[ g_{n}(x) ]d[x] ] = int[ g(x) ]d[x]

Teorema:

Sea ( u(a,b,n) = min{f(x)+(-1)·(a/n),f(x)+(-1)·(b/n)} & & |A_{oo}(a,b)| = 1 ) ==>

Sea s_{n}(x) = u(a,b,n)·|A_{n}(a,b)| ==>

lim[n = oo][ int[x = a]-[b][ s_{n}(x) ]d[x] ] = int[x = a]-[b][ f(x) ]d[x]

Demostración:

u(a,b,n)·|A_{n}(a,b)| [< u(a,b,n)·|A_{n+1}(a,b)| [< ...

... u(a·(n/(n+1)),b·(n/(n+1)),n)·|A_{n+1}(a,b)| = u(a,b,n+1)·|A_{n+1}(a,b)|

Si [ (MP) Teorema de convergencia monótona ] ==>

lim[n = oo][ int[x = a]-[b][ s_{n}(x) ]d[x] ] = int[x = a]-[b][ s(x) ]d[x] = int[x = a]-[b][ f(x) ]d[x]

Teorema:

Sea ( v(a,b,n) = max{f(x)+(-1)·(a/n),f(x)+(-1)·(b/n)} & |B_{(-oo)}(a,b)| = 1 ) ==>

Sea S_{n}(a,b) = v(a,b,n)·|B_{(-n)}(a,b)| ==>

lim[n = oo][ int[x = a]-[b][ S_{n}(x) ]d[x] ] = int[x = a]-[b][ f(x) ]d[x]

Demostración:

v(a,b,n)·|B_{(-n)}(a,b)| >] v(a,b,n)·|B_{(-1)·(n+1)}(a,b)| >] ...

... v(a·(n/(n+1)),b·(n/(n+1)),n)·|B_{(-1)·(n+1)}(a,b)| = v(a,b,n+1)·|B_{(-1)·(n+1)}(a,b)|

Si [ (MP) Teorema de convergencia monótona ] ==>

lim[n = oo][ int[x = a]-[b][ S_{n}(x) ]d[x] ] = int[x = a]-[b][ S(x) ]d[x] = int[x = a]-[b][ f(x) ]d[x]

Teorema:

Sea m >] n ==>

Sea s_{n}(a,b) = ( f(x)+(-1)·(b/n) )·( m+(-n) ) ==>

lim[n = oo][ int[x = a]-[b][ s_{n}(x) ]d[x] ] = F(b)+(-1)·F(a)

Sea (-m) [< (-n) ==>

Sea S_{n}(a,b) = ( f(x)+(-1)·(a/n) )·( n+(-m) ) ==>

lim[n = oo][ int[x = a]-[b][ S_{n}(x) ]d[x] ] = F(b)+(-1)·F(a)

Demostración:

Sea m >] n+1 >] n ==>

(-n) >] (-n)+(-1) = (-1)·(n+1)

(-n)+m >] (-1)·(n+1)+m

|A_{n+1}(a,b)| = |A_{n}(a,b)|+(-1)

s_{n}(a,b) = ( f(x)+(-1)·(b/n) )·( m+(-n) ) [< ( f(x)+(-1)·(b/(n+1)) )·( m+(-1)·(n+1) ) = s_{n+1}(0,a)

Sea (-m) [< (-1)·(n+1) [< (-n) ==>

n [< n+1

n+(-m) [< n+1+m

|B_{n+1}(a,b)| = |B_{n}(a,b)|+1

S_{n}(a,b) = ( f(x)+(-1)·(a/n) )·( n+(-m) ) >] ( f(x)+(-1)·(a/(n+1)) )·( (n+1)+(-m) ) = S_{n+1}(a,b)

Teorema:

Sea ( c = b+(-a) >] 0 ==> sig(b+(-a)) = 1 ) ==>

Sea s_{n}(a,b) = ( f(x)+(-1)·(b/n) )·( (n+1)+(-n) )·sig(c) ==>

lim[n = oo][ int[x = a]-[b][ s_{n}(x) ]d[x] ] = F(b)+(-1)·F(a)

Sea ( (-c) = a+(-b) [< 0 ==> sig(a+(-b)) = (-1) ) ==>

Sea S_{n}(a,b) = ( f(x)+(-1)·(a/n) )·( (n+1)+(-n) )·sig(-c) ==>

lim[n = oo][ int[x = a]-[b][ S_{n}(x) ]d[x] ] = F(b)+(-1)·F(a)

Teorema:

Sea s_{n}(a,b) = ( f(x)+(-1)·(b/n) )·( (n+1)+(-n) )·( 1+(-1)·(1/n) ) ==>

lim[n = oo][ int[x = a]-[b][ s_{n}(x) ]d[x] ] = F(b)+(-1)·F(a)

Sea S_{n}(a,b) = ( f(x)+(-1)·(a/n) )·( (n+1)+(-n) )·( (-1)+(1/n) ) ==>

lim[n = oo][ int[x = a]-[b][ S_{n}(x) ]d[x] ] = F(b)+(-1)·F(a)

Demostración:

Sea |¬A_{n}(a,b)| = (1/n) ==>

|A_{n+1}(a,b)| = 1+(-1)·|¬A_{n}(a,b)|·( n/(n+1) )

1+(-1)·( 1/(n+1) ) = 1+(-1)·(1/n)·( n/(n+1) )

s_{n}(a,b) = ( f(x)+(-1)·(b/n) )·( (n+1)+(-n) )·( 1+(-1)·(1/n) ) [< ...

... ( f(x)+(-1)·(b/(n+1)) )·( (n+1)+(-n) )·( 1+(-1)·(1/(n+1)) ) = s_{n+1}(a,b)

Sea |¬B_{(-n)}(a,b)| = (1/n) ==>

|B_{(-1)·(n+1)}(a,b)| = (-1)+|¬B_{(-n)}(a,b)|·( n/(n+1) )

(-1)+( 1/(n+1) ) = (-1)+(1/n)·( n/(n+1) )

S_{n}(a,b) = ( f(x)+(-1)·(a/n) )·( (n+1)+(-n) )·( (-1)+(1/n) ) >] ...

... ( f(x)+(-1)·(a/(n+1)) )·( (n+1)+(-n) )·( (-1)+(1/(n+1)) ) = S_{n+1}(a,b)

Teorema: [ de la función de doble factorial exponencial ]

[1] Sea sum[k = 2p]-[n][ (1/(k+2)!!)·z^{k+2} ] = [!e!]-(z) ==>

Sea s_{n}(a,b) = ( f(x)+(-1)·(b/n) )·...

... ( (n+1)+(-n) )·sum[k = 2p]-[n][ (-1)^{k}·(1/(k+2)!!)·z^{k+2} ]·( 1/[!e!]-(z) ) ==>

lim[n = oo][ int[x = a]-[b][ s_{n}(x) ]d[x] ] = F(b)+(-1)·F(a)

[2] Sea sum[k = 2p+1]-[n][ (1/(k+1)!!)·(-z)^{k+1} ] = [!e!]-(-z) ==>

Sea S_{n}(a,b) = ( f(x)+(-1)·(a/n) )·...

... ( (n+1)+(-n) )·sum[k = 2p+1]-[n][ (-1)^{k}·(1/(k+1)!!)·(-z)^{k+1} ]·( 1/[!e!]-(-z) ) ==>

lim[n = oo][ int[x = a]-[b][ S_{n}(x) ]d[x] ] = F(b)+(-1)·F(a)

Demostración:

[1] |A_{2k+2}(a,b)| = |A_{2k}(a,b)|+(-1)^{n}·(1/(n+2)!!)·z^{n+2}·( 1/[!e!]-(z) )

Si n = 2p ==>

z^{n+2} >] 0

(-1)^{n}·(1/(n+2)!!)·z^{n+2}·( 1/[!e!]-(z) ) >] 0

s_{n}(a,b) = ( f(x)+(-1)·(b/n) )·((n+1)+(-n))·...

... sum[k = 2p]-[n][ (-1)^{k}·(1/(k+2)!!)·z^{k+2} ]·( 1/[!e!]-(z) ) [< ...

... ( f(x)+(-1)·(b/(n+2)) )·( ((n+2)+1)+(-1)·(n+2) )·...

... sum[k = 2p]-[n+2][ (-1)^{k}·(1/(k+2)!!)·z^{k+2} ]·( 1/[!e!]-(z) ) = s_{n+2}(a,b)

[2] |B_{(-1)·(2k+3)}(a,b)| = |B_{(-1)·(2k+1)}(a,b)|+(-1)^{n}·(1/(n+1)!!)·(-z)^{n+1}·( 1/[!e!]-(-z) )

Si n = 2p+1 ==>

(-z)^{n+1} >] 0

(-1)^{n}·(1/(n+1)!!)·(-z)^{n+1}·( 1/[!e!]-(-z) ) [< 0

S_{n}(a,b) = ( f(x)+(-1)·(a/n) )·((n+1)+(-n))·...

... sum[k = 2p+1]-[n][ (-1)^{k}·(1/(k+1)!!)·(-z)^{k+1} ]·( 1/[!e!]-(-z) ) >] ...

... ( f(x)+(-1)·(a/(n+2)) )·( ((n+2)+1)+(-1)·(n+2) )·...

... sum[k = 2p+1]-[n+2][ (-1)^{k}·(1/(k+1)!!)·(-z)^{k+1} ]·(1/[!e!]-(-z)) = S_{n+2}(a,b)

Teorema:

d_{x}[ [!e!]-(x) ] = [!e!]-(x)

d_{x}[ [!e!]-(-x) ] = (-1)·[!e!]-(-x)

Demostración:

Sea k = m+2 ==>

d_{x}[ [!e!]-(x) ] = sum[k = 2p]-[oo][ (1/k!!)·x^{k} ] = ...

... sum[m = 2p]-[oo][ (1/(m+2)!!)·x^{m+2} ] = [!e!]-(x)

d_{x}[ [!e!]-(-x) ] = sum[k = 2p+1]-[oo][ (-1)·(1/(k+(-1))!!)·(-x)^{k+(-1)} ] = ...

... (-1)·sum[k = 2p+1]-[oo][ (1/(k+(-1))!!)·(-x)^{k+(-1)} ] = ...

... (-1)·sum[m = 2p+1]-[oo][ (1/(m+1)!!)·(-x)^{m+1} ] = (-1)·[!e!]-(-x)

 

Teorema:

Sea s_{n}(a,b) = ( f(x)+(-1)·(b/n) )·( (n+1)+(-n) )·...

... sum[k = 2p]-[n][ (-1)^{k}·(1/( (1/2)·(k+2) )!)·z^{k+2} ]·e^{(-1)·z^{2}} ) ==>

lim[n = oo][ int[x = a]-[b][ s_{n}(x) ]d[x] ] = F(b)+(-1)·F(a)

Sea S_{n}(a,b) = ( f(x)+(-1)·(a/n) )·( (n+1)+(-n) )·...

... sum[k = 2p+1]-[n][ (-1)^{k}·(1/( (1/2)·(k+1) )!)·(-z)^{k+1} ]·e^{(-1)·z^{2}} ) ==>

lim[n = oo][ int[x = a]-[b][ S_{n}(x) ]d[x] ] = F(b)+(-1)·F(a)

Teorema:

Sea < x€Q ==> f(x) = 1 & x€I ==> f(x) = 0 > ==> int[x = 0]-[1][ f(x) ]d[x] = 1

Demostración:

Sea |M_{n}(0,1)| = oo·|B_{n}(0,1)| ==>

s_{n}(0,1) = ( 1+(-1)·(1/n) )·|A_{n}(0,1)| = [ x ]_{x = 0}^{x = 1} = 1

S_{n}(0,1) = ( 0+(-1)·(0/n) )·|M_{n}(0,1)| = [ x ]_{x = 0}^{x = 1} = 1

Teorema:

Sea < x€Q ==> f(x) = x & x€I ==> f(x) = 1+(-x) > ==> int[x = 0]-[1][ f(x) ]d[x] = (1/2)

Demostración: 

s_{n}(0,1) = ( (1/2)+(-1)·(1/n) )·|A_{n}(0,1)| = [ (1/2)·x ]_{x = 0}^{x = 1} = (1/2)

S_{n}(0,1) = ( (1+(-1)·(1/2))+(-1)·(0/n) )·|B_{n}(0,1)| = [ (1/2)·x ]_{x = 0}^{x = 1} = (1/2)

Teorema:

[ {a_{1},...,a_{n}} ] ==> [ {f(a_{1}),...,f(a_{n})} ] ==> [ {g(f(a_{1})),...,g(f(a_{n})} ] es especie

Demostración:

g(f(p)) = g(f(q))

f(p) = f(q)

p = q

Teorema:

[ < a_{1},...,a_{n} > ] ==> [ < f(a_{1}),...,f(a_{n}) > ] ==> [ < g(f(a_{1})),...,g(f(a_{n}) > ] es especie

Demostración:

g(f(p)) = g(f(q))

f(p) = f(q)

p = q

Definición: [ de serie aritmética de una especie ]

Sea A una especie ==>

F(x) = sum[n = 0]-[oo][ #A(n)·x^{n} ]

Definición: [ de isomorfismo de especie ]

Sean A & B especies ==>

A =[#]= B <==> sum[n = 0]-[oo][ #A(n)·x^{n} ] = sum[n = 0]-[oo][ #B(n)·x^{n} ]

Teorema:

#A(n) = #A(n)

#A(n) = #B(n) <==> #B(n) = #A(n)

Si ( #A(n) = #B(n) & #B(n) = #C(n) ) ==> #A(n) = #C(n) 

Teorema:

A(n)+[ M ] =[#]= A(n+1)

Demostración:

sum[n = 0]-[oo][ ( #A(n)+#[ M ] )·x^{n} ] = ...

... sum[n = 0]-[oo][ (n+1)·x^{n} ] = sum[n = 0]-[oo][ #A(n+1)·x^{n} ]

Teorema:

Sea A una especie de orden inferior a A(n) ==>

[EB][ B = A [& || &] A(n) & B [<< A(n) ]

Demostración:

Se define B = A [& || &] A(n) ==>

B = B [&] A(n)

B [<< A(n)

Teorema:

Sea A una especie de orden inferior a N(n) ==>

[EB][ B = A [& | &] N(n) & B [<< N(n) ]

Demostración:

Se define B = A [& | &] N(n) ==>

B = B [&] N(n) 

B [<< N(n)

Definición: [ de serie geométrica de una especie ]

Sea A(p,n) una especie geométrica ==>

F(x) = sum[n = 0]-[oo][ p^{#A(n)}·x^{n} ]

Teorema:

Sea A(p,n) & B(q,n) dos especies geométricas & a = mcd{p,q} ==>

[EM][ M es especie geométrica & M = mcd{A(p,n),B(q,n)} & F(1) = ( a/(a+(-1)) ) ]

Demostración:

Se define M(mcd{p,q},n) = mcd{A(p,n),B(q,n)} ==>

M(a,n) = M(mcd{p,q},n)

F(1) = sum[n = 0]-[oo][ (1/a)^{n} ] = ( a/(a+(-1)) )

Teorema:

Sea A(p,n) & B(q,n) dos especies geométricas & b = mcm{p,q} ==>

[EW][ W es especie geométrica & W = mcm{A(p,n),B(q,n)} & F(1) = ( b/(b+(-1)) ) ]

Demostración:

Se define W(mcm{p,q},n) = mcm{A(p,n),B(q,n)} ==>

W(b,n) = W(mcm{p,q},n)

F(1) = sum[n = 0]-[oo][ (1/b)^{n} ] = ( b/(b+(-1)) )

Definición:

P_{k}( {a_{1},...,a_{n}} ) es especie

F(x) = sum[n = 0]-[oo][ [ n // k ]·x^{n} ]

Teorema:

Sea f(n) = [ n // n+(-1) ] ==>

[EA][ A = [ 2 // 2 ] [& || &] [ 3 // 2 ] & A [<< [ 3 // 2 ] ]

[EB][ B = A [& || &] [ 4 // 3 ] & B [<< [ 4 // 3 ] ]

Demostración:

A = [ {a,b} ]

B = [ {a,b,c},{a,b,d} ]

Arte:

[Ex][ Si F(x) = sum[n = 1]-[oo][ [ n // n+(-1) ]·x^{n} ] ==> d_{x}[F(x)] = ( 1/(1+(-x)) ) ]

Exposición:

x = 0

d_{x}[F(x)] = sum[n = 1]-[oo][ n^{2}·x^{n+(-1)} ] = sum[n = 1]-[oo][ 2n·x^{n+(-1)} ] = ...

... sum[n = 1]-[oo][ (n+n+(1/2)+(-1)·(1/2))·x^{n+(-1)} ] = ...

... sum[n = 1]-[oo][ (n+(-n)+(1/2)+(1/2))·x^{n+(-1)} ] = sum[n = 1]-[oo][ x^{n+(-1)} ] = ( 1/(1+(-x)) )

Teorema:

Sea f(n) = [ 2n // n ] ==>

[EA][ A = [ 2 // 2 ] [& || &] [ 4 // 2 ] & A [<< [ 4 // 2 ] ]

[EB][ B = A [& || &] [ 6 // 3 ] & B [<< [ 6 // 3 ] ]

Demostración:

A = [ {a,b} ]

B = [ {a,b,c},{a,b,d},{a,b,e},{a,b,f} ]

Arte:

[Ex][ Si F(x) = sum[n = 0]-[oo][ [ 2n // n ]·x^{n} ] ==> F(x) = ( 1/(1+(-x)) ) ]

Exposición:

x = 0

F(x) = sum[n = 0]-[oo][ (2n)!(1/n!)·(1/n!)·x^{n} ] = sum[n = 0]-[oo][ (2n)·(n!/n!)·x^{n} ] = ...

... sum[n = 0]-[oo][ (2n)!·x^{n} ] = sum[n = 0]-[oo][ (n+n)!·x^{n} ] = ...

... sum[n = 0]-[oo][ (n+(-n))!·x^{n} ] = sum[n = 0]-[oo][ 0!·x^{n} ] = ...

... sum[n = 0]-[oo][ x^{n} ] = ( 1/(1+(-x)) )

Definición:

A(n) = { B : i+j = n & B = {a}^{i} [x] {b}^{j} } es especie

F(x) = sum[n = 2]-[oo][ sum[i+j = n][ a^{i}·b^{j} ]·x^{n} ]

Teorema:

Sea A(4) = [ < a,a,a,b >,< a,a,b,b >,< b,b,b,a > ] ==>

[EM][ M = [ < a,a,b > ] [& || &] A(4) & M [<< A(4) ]

[EW][ W = [ < b,b,a > ] [& || &] A(4) & W [<< A(4) ]

Demostración:

M = [ < a,a,a,b >,< a,a,b,b > ]

W = [ < b,b,b,a >,< a,a,b,b > ]

Teorema:

M+W = ab·(a+b)^{2}

Demostración:

( ba^{3}+a^{2}·b^{2} )+( a^{2}·b^{2}+ab^{3} ) = ...

... ba^{3}+2a^{2}·b^{2}+ab^{3} = ab·( a^{2}+2ab+b^{2} )

Arte:

(1/2)·M+(1/2)·W != 2ab·(a+b)^{2}

Exposición:

( (1/2)·ba^{3}+(1/2)·a^{2}·b^{2} )+( (1/2)·a^{2}·b^{2}+(1/2)·ab^{3} ) = ...

... ( 2ba^{3}+2a^{2}·b^{2} )+( 2a^{2}·b^{2}+2ab^{3} ) = ...

... 2ba^{3}+4a^{2}·b^{2}+2ab^{3} = 2ab·( a^{2}+2ab+b^{2} )

Teorema:

Si F(x) = sum[n = 2]-[oo][ sum[i+j = n][ a^{i}·b^{j} ]·x^{n} ] ==> ...

... F(1) = ab·( 1/(1+(-a)) )·( 1/(1+(-b)) )

Demostración:

F(1) = ( 1/(1+(-a)) )·( 1/(1+(-b)) )+(-1)·( 1/(1+(-a)) )+(-1)·( 1/(1+(-b)) )+1

Definición:

A(n) = { < f(1),...,f(n) > } es especie

F(x) = sum[n = 1]-[oo][ n!·x^{n} ]

Teorema:

Sea ( A = { < f(1),f(2) > : [Ek][ f(k) = k ] } & B = { < f(1),f(2) > : [Ak][ f(k) != k ] } ) ==> ...

... [EM][ M = A [& || &] A(3) & M [<< A(3) ]

... [EW][ W = B [& || &] A(3) & W [<< A(3) ]

Demostración:

M = [ < 1,2,3 > ]

W = [ < 2,1,3 > ]

Teorema:

Sea ( A = { < f(1),f(2) > : [Ek][ f(k) = k ] } & B = { < f(1),f(2) > : [Ak][ f(k) != k ] } ) ==> ...

... [EM][ M = A [& || &] A(4) & M [<< A(4) ]

... [EW][ W = B [& || &] A(4) & W [<< A(4) ]

Demostración:

M = [ < 1,2,3,4 >,< 1,2,4,3 > ]

W = [ < 2,1,3,4 >,< 2,1,4,3 > ]

Arte:

Sea [ < f(1),...,f(n) > ] ==> d_{x}[F(x)] != sum[n = 1]-[oo][ n!·x^{n} ]

Exposición:

x = 0

d_{x}[F(x)] = sum[n = 1]-[oo][ n!·n·x^{n+(-1)} ] = sum[n = 0]-[oo][ (n+1)!·(n+1)·x^{n} ] = ...

... sum[n = 0]-[oo][ (n+1)!·( (n+1)+(1/2)+(-1)·(1/2) )·x^{n} ] = ...

... sum[n = 0]-[oo][ (n+1)!·( (n+1)+(1/2)+(1/2) )·x^{n} ] = sum[n = 0]-[oo][ (n+1)!·(n+2)·x^{n} ] = ...

... sum[n = 0]-[oo][ (n+2)!·x^{n} ] = sum[n = 0]-[oo][ ( n+(1+1) )!·x^{n} ] = ...

... sum[n = 0]-[oo][ ( n+(1+(-1)) )!·x^{n} ] = sum[n = 0]-[oo][ n!·x^{n} ]

Definición: [ de octopus geométrico ]

A = [ {p},...,{p^{n}} ]-[ \ ]-[ {p^{j}} ]

F(A,x) = sum[n = 1]-[oo][ p^{n+(-1)}·x^{n} ]

Teorema:

Sea ( A = [ {(1/6)},...,{(1/6)^{n}} ]-[ \ ]-[ {(1/6)^{j}} ] & ...

... B = [ {(1/10)},...,{(1/10)^{n}} ]-[ \ ]-[ {(1/10)^{j}} ] ) ==>

[EM][ M = mcd{A,B} & F(M,1) = ? ]

[EW][ W = mcm{A,B} & F(W,1) = ? ]

Demostración:

6 = 2·3 & 10 = 2·5

M = [ {(1/2)},...,{(1/2)^{n}} ]-[ \ ]-[ {(1/2)^{j}} ]

F(M,1) = 2

W = [ {(1/30)},...,{(1/30)^{n}} ]-[ \ ]-[ {(1/30)^{j}} ]

F(W,1) = (30/29)

Teorema:

Sea [ || ]-[j = 1]-[n][ [ {p},...,{p^{n}} ]-[ \ ]-[ {p^{j}} ] ] ==>

Si F(x) = sum[n = 1]-[oo][ p^{n^{2}+(-n)}·x^{n} ] ==> F(1) = ( 1/(1+(-1)·p^{n}) )

Demostración:

F(1) = sum[n = 1]-[oo][ p^{n^{2}+(-n)} ] = ...

... sum[n = 1]-[oo][ ( p^{n} )^{n+(-1)} ] = ( 1/(1+(-1)·p^{n}) )

Arte:

Si F(x) = sum[n = 1]-[oo][ p^{n+(-1)}·x^{n} ] ==> F(1) != p·( 1/(1+(-p)) )

Exposición:

F(1) = sum[n = 1]-[oo][ p^{n+(-1)} ] = sum[n = 1]-[oo][ p^{n+(-1)·( (1/2)+(1/2) )} ] = ...

... sum[n = 1]-[oo][ p^{n+(-1)·(1/2)+(1/2)} ] = sum[n = 1]-[oo][ p^{n} ] = ...

... p·sum[n = 1]-[oo][ p^{n+(-1)} ] = p·( 1/(1+(-p)) )

Definición: [ de octopus aritmético ]

N(n) = [ {1},...,{n} ]

F(x) = sum[n = 1]-[oo][ nx^{n} ]

Teorema:

Sea ( A = [ {2},{4} ] & B = [ {2},{3} ] ) ==> ...

... [EM][ M = A [& | &] A(8) & M [<< A(8) ]

... [EW][ W = B [& | &] A(12) & W [<< A(12) ]

Demostración:

M = [ {4},{8} ]

W = [ {6},{12} ]

Teorema:

Sea A(n+1) = [ {a_{1},...,a_{n}} ]-[ {a_{1}},...,{a_{n}} ] ==>

[EM][ M = N(n) x A(n+1) ]

M =[#]= 2·sum[k = 1]-[n][ [ {a_{1}},...,{a_{k}} ] ]

Demostración:

M = [ < {1},{a_{1},...,a_{n}} >,...,< {n},{a_{1},...,a_{n}} > ]-...

... [ < {1},{a_{1}} >,...,< {1},{a_{n}} >,...,< {n},{a_{1}} >,...,< {n},{a_{n}} > ]

F(M,x) = sum[n = 1]-[oo][ n·(n+1)·x^{n} ] = sum[n = 1]-[oo][ (n^{2}+n)·x^{n} ]

(1/2)·n·(n+1)+(n+1) = (1/2)·( n·(n+1)+2·(n+1) ) = (1/2)·(n+2)·(n+1) = (1/2)·(n+1)·(n+2)

Teorema:

Sea A(n) = [ {a_{1}},...,{a_{n}} ] ==>

[EM][ M = N(n) x A(n) ]

M =[#]= sum[k = 1]-[n][ [ {a_{1}},...,{a_{2k+(-1)}} ] ]

Demostración:

M = [ < {1},{a_{1}} >,...,< {1},{a_{n}} >,...,< {n},{a_{1}} >,...,< {n},{a_{n}} > ]

F(M,x) = sum[n = 1]-[oo][ n·n·x^{n} ] = sum[n = 1]-[oo][ n^{2}·x^{n} ]

n^{2}+(2n+1) = (n+1)^{2}

Teorema:

Sea A(n) = [ {a_{1}},...,{a_{n}} ] ==>

[EM][ M = ( N(n) x A(n) )+( A(n) x N(n) ) ]

M =[#]= (1/2)·sum[k = 1]-[n][ [ {a_{4}},...,{a_{8k+(-4)}} ] ]

Demostración:

M = ...

... [ < {1},{a_{1}} >,...,< {1},{a_{n}} >,...,< {n},{a_{1}} >,...,< {n},{a_{n}} > ] ...

... +...

... [ < {a_{1}},{1} >,...,< {a_{n}},{1} >,...,< {a_{1}},{n} >,...,< {a_{n}},{n} > ]

F(M,x) = sum[n = 1]-[oo][ (n^{2}+n^{2})·x^{n} ] = sum[n = 1]-[oo][ 2n^{2}·x^{n} ]

4n^{2}+(8n+4) = (2n+2)^{2} = 4·(n+1)^{2}

Teorema:

Sea A(p,n) = [ {(1/p)},...,{(1/p)^{n}} ] ==>

[EM][ M = N(n) [o] A(p,n) ]

F(1) = p·( 1/(p+(-1))^{2} )

Demostración:

M(n) = [ < {1},{(1/p)^{n}} >,...,< {n},{(1/p)^{n}} > ] ...

Teorema:

Sea A(p,n) = [ {(1/p)},...,{(1/p)^{n}} ] ==>

[EM][ M = ( N(n) x N(n) ) [o] A(p,n) ]

F(1) = p·( 1/(p+(-1))^{3} )+p·( 1/(p+(-1))^{2} )

Demostración:

M(n) = [ < {1}x{1},{(1/p)}^{n} >,...,< {1}x{n},{(1/p)^{n}} >,...(n)...,

... < {n}x{1},{(1/p)}^{n} >,...,< {n}x{n},{(1/p)^{n}} > ]

Ley:

La enfermedad de creer-se señor del prójimo,

no impide creer en condenación

porque las voces de la mente no pueden salir de la mente,

en creer que hay condenación del satélite.

Deducción:

La enfermedad de creer-se señor del prójimo,

impide creer en condenación

aunque quizás las voces de la mente no pueden salir de la mente,

en creer que hay condenación del satélite.

Teorema:

Sea f_{n}(x) = (1/n)·x^{2p+(-1)} ==> ...

... ( f_{n}(x) es integrable en R & lim[n = oo][ int[ f_{n}(x) ]d[x] ] = (1/oo)·(1/(2p))·x^{2p}

Demostración:

Sea x > 0 ==>

(1/n) >] (1/(n+1))

f_{n}(x) = (1/n)·x^{2p+(-1)} >] (1/(n+1))·x^{2p+(-1)} = f_{n+1}(x)

Si [ (MP) Teorema de convergencia monótona ] ==>

f_{n}(x) es integrable

Sea x < 0 ==>

(1/n) >] (1/(n+1))

f_{n}(x) = (1/n)·x^{2p+(-1)} [< (1/(n+1))·x^{2p+(-1)} = f_{n+1}(x)

Si [ (MP) Teorema de convergencia monótona ] ==>

f_{n}(x) es integrable

Sea x = 0 ==>

f_{n}(x) = (1/n)·x^{2p+(-1)} >] (1/oo)·x^{2p+(-1)} = f(x)

Si [ (MP) Teorema de convergencia dominada ] ==>

f_{n}(x) es integrable

Sea x = (-0) ==>

f_{n}(x) = (1/n)·x^{2p+(-1)} [< (1/oo)·x^{2p+(-1)} = f(x)

Si [ (MP) Teorema de convergencia dominada ] ==>

f_{n}(x) es integrable

lim[n = oo][ int[ f_{n}(x) ]d[x] ] = int[ f(x) ]d[x] = ...

... int[ (1/oo)·x^{2p+(-1)} ]d[x] = (1/oo)·(1/(2p))·x^{2p}

Teorema:

Sea f_{n}(x) = nx^{2p+1} ==> ...

... ( f_{n}(x) es integrable en R & lim[n = oo][ int[ f_{n}(x) ]d[x] ] = oo·(1/(2p+2))·x^{2p+2} )

Demostración:

Sea x > 0 ==>

n [< n+1

f_{n}(x) = nx^{2p+1} [< (n+1)·x^{2p+1} = f_{n+1}(x)

Si [ (MP) Teorema de convergencia monótona ] ==>

f_{n}(x) es integrable

Sea x < 0 ==>

n [< n+1

f_{n}(x) = nx^{2p+1} >] (n+1)·x^{2p+1} = f_{n+1}(x)

Si [ (MP) Teorema de convergencia monótona ] ==>

f_{n}(x) es integrable

Sea x = 0 ==>

f_{n}(x) = nx^{2p+1} [< oo·x^{2p+1} = f(x)

Si [ (MP) Teorema de convergencia dominada ] ==>

f_{n}(x) es integrable en x = 0

Sea x = (-0) ==>

f_{n}(x) = nx^{2p+1} >] oo·x^{2p+1} = f(x)

Si [ (MP) Teorema de convergencia dominada ] ==>

f_{n}(x) es integrable en x = (-0)

lim[n = oo][ int[ f_{n}(x) ]d[x] ] = int[ f(x) ]d[x] = int[ oo·x^{2p+1} ]d[x] = oo·(1/(2p+2))·x^{2p+2}

Exámenes de Análisis matemático 5:

Teorema:

Sea f_{n}(x) = n·ln(x) ==> ...

... ( f_{n}(x) es integrable en x > 0 & lim[n = oo][ int[ f_{n}(x) ]d[x] ] = oo·( ln(x)·x+(-x) )

Teorema:

Sea f_{n}(x) = n·( e^{x}+(-1) ) ==> ...

... ( f_{n}(x) es integrable en R & lim[n = oo][ int[ f_{n}(x) ]d[x] ] = oo·( e^{x}+(-x) )

Diferencial exterior de producto escalar nulo.

Teorema:

int-int[ x^{n}·d[x]d[y]+y^{n}·d[y]d[x] ] = (1/(n+1))·x^{n}·y^{n}·( y+x )

x^{n}·( d[x] & d[y] )+y^{n}·( d[y] & d[x] ) = x+(-y)

Teorema:

int-int[ x^{n}·d[x]d[y]+y^{n+k}·d[y]d[x] ] = x^{n}·y^{n}·( (1/(n+1))·y+(1/(n+k+1))·y^{k+1}·x )

x^{n}·( d[x] & d[y] )+y^{n+k}·( d[y] & d[x] ) = (n+1)·x+(-1)·(n+k+1)·(1/y)^{k}

Teorema:

int-int[ e^{nx}·d[x]d[y]+y^{n}·d[y]d[x] ] = (1/n)·e^{nx}·y+(1/(n+1))·y^{n+1}·x

e^{nx}·( d[x] & d[y] )+y^{n}·( d[y] & d[x] ) = nxe^{(-1)·nx}+(-1)·(n+1)·(1/y)^{n}

Teorema:

int-int[ yx^{n}·d[x]d[y]+xy^{n}·d[y]d[x] ] = (1/2)·(1/(n+1))·x^{n}·y^{n}·( y^{2}+x^{2} )

yx^{n}·( d[x] & d[y] )+xy^{n}·( d[y] & d[x] ) = x^{2}+(-1)·y^{2}

Matemáticas y Física:

Análisis matemático 3:

[%] Derivadas parciales.

[%] Optimización de LaGrange.

Trayectorias sobre superficies.

Análisis matemático 4:

[%] Integrales múltiples.

[%] Integrales de camino ( d[x] & d[yz] )

Diferenciales exteriores.

Matemáticas:

Análisis matemático 5:

Sucesiones de funciones integrables.

Integral de Medida = 1.

Series de Fourier.

Análisis matemático 6:

Transformada integral exponencial.

Integrales por el Método de Euler.

Series de Laurent.

Definición:

a_{0} = 0

a_{k} = (1/pi·i)·int[0]-[2pi·i][ f(x)·cosh(kx) ]d[x] 

b_{k} = (-1)·(1/pi·i)·int[0]-[2pi·i][ f(x)·sinh(kx) ]d[x]

cosh(2w) = ( cosh(w) )^{2}+( sinh(w) )^{2}

Teorema:

x^{3} = (-12)·( sinh(x)+(1/3)^{3}·sinh(3x)...+(1/(2n+1))^{3}·sinh((2n+1)·x) )

sum[k = 0]-[oo][ (-1)^{k}·( 1/(2k+1) )^{3} ] = (1/96)·pi^{3}

Demostración:

0 [< x [< 2pi·i = 0

( (pi/2)·i )^{3} = (-1)·(1/8)·pi^{3}·i

Teorema:

x^{5} = 160·pi^{2}·( sinh(x)+(1/3)^{3}·sinh(3x)...+(1/(2n+1))^{3}·sinh((2n+1)·x) )+...

... (-240)·( sinh(x)+(1/3)^{5}·sinh(3x)...+(1/(2n+1))^{5}·sinh((2n+1)·x) )

sum[k = 0]-[oo][ (-1)^{k}·( 1/(2k+1) )^{5} ] = (157/23,040)·pi^{5}

Demostración:

0 [< x [< 2pi·i = 0

( (pi/2)·i )^{5} = (1/32)·pi^{5}·i

(-1)·(1/pi·i)·( (1/(2k+1))^{3}·5·4·8pi^{3}·(-i)+(1/(2k+1))^{5}·5·4·3·2·2pi·i )

Trayectorias en superficies:

Definición:

[As][Ad][ ( s > 0 & d > 0 ) ==> ( Si |h(kx)+(-y)| < d ==> | M(x,h(kx))+(-1)·F(x,y) | < s ) ]

Teorema:

Si ( lim[y = h(kx)][ F(x,y) ] = M(x,h(kx)) & lim[y = h(kx)][ F(x,y) ] = N(x,h(kx)) ) ==> ...

... M(x,h(kx)) = N(x,h(kx))

Demostración:

Sea s > 0 ==>

| M(x,h(kx))+(-1)·N(x,h(kx)) | = | M(x,h(kx))+(-1)·F(x,y)+F(x,y)+(-1)·N(x,h(kx)) | [< ...

... | M(x,h(kx))+(-1)·F(x,y) |+| F(x,y)+(-1)·N(x,h(kx)) | < (s/2)+(s/2) = s

Teorema:

Si ( lim[y = h(kx)][ F(x,y) ] = M(x,h(kx)) & lim[y = h(kx)][ G(x,y) ] = N(x,h(kx)) ) ==> ...

... lim[y = h(kx)][ F(x,y)+G(x,y) ] = M(x,h(kx))+N(x,h(kx))

Teorema:

Si lim[y = h(kx)][ F(x,y) ] = M(x,h(kx)) ==> lim[y = h(kx)][ w·F(x,y) ] = w·M(x,h(kx))

Teorema:

Sea lim[y = h(kx)][ F(x,y) ] = M(x,h(kx)) ==> F(x,y) está acotada

Demostración:

h(kx) = y = c

x = (1/k)·h^{o(-1)}(c)

|F(x,y)| [< | F(x,y)+(-1)·M(x,h(kx)) |+|M(x,h(kx))|

Se define M = max{s,M(x,h(kx))} ==>

|F(x,y)| [< M

Teorema:

Sea lim[y = h(kx)][ F(x,y) ] = M(x,h(kx)) ==> M(x,h(kx)) está acotada

Demostración:

h(kx) = y = c

x = (1/k)·h^{o(-1)}(c)

|M(x,h(kx))| [< | M(x,h(kx))+(-1)·F(x,y) |+|F(x,y)|

Se define M = max{s,F(x,y)} ==>

|M(x,h(kx))| [< M

Teorema:

Si ( lim[y = h(kx)][ F(x,y) ] = M(x,h(kx)) & lim[y = h(kx)][ G(x,y) ] = N(x,h(kx)) ) ==> ...

... lim[y = h(kx)][ F(x,y)·G(x,y) ] = M(x,h(kx))·N(x,h(kx))

Demostración:

Sea s > 0 ==>

| M(x,h(kx))·N(x,h(kx))+(-1)·F(x,y)·G(x,y) | = ...

... | M(x,h(kx))·N(x,h(kx))+(-1)·F(x,y)·N(x,h(kx))+F(x,y)·N(x,h(kx))+(-1)·F(x,y)·G(x,y) | [< ...

... | M(x,h(kx))+(-1)·F(x,y) |·|N(x,h(kx))|+|F(x,y)|·| N(x,h(kx))+(-1)·G(x,y) | < (s/2)+(s/2) = s

Teorema:

lim[y = kx][ ( (x^{n+1}+y^{n+1})/(x+y) ) ] = x^{n}·k^{[n+1:1]+(-1)·[1:1]}

Demostración:

( (x^{n+1}+(kx)^{n+1})/(x+kx) ) = ( (x^{n+1}+k^{n+1}·x^{n+1})/(x+kx) ) = ...

... ( ((1+k^{n+1})·x^{n+1})/((1+k)·x) ) = ( (1+k^{n+1})/(1+k) )·x^{n}

Teorema:

lim[y = ln(kx)][ ( (x^{n+1}+e^{(n+1)·y})/(x+e^{y}) ) ] = x^{n}·k^{[n+1:1]+(-1)·[1:1]}

Teorema:

lim[y = e^{kx}][ ( ln(e^{nx}+y)/x ) ] = k+[n+(-k):1]

topología y especies-combinatorias y filosofía y medicina y dualogía y análisis-matemático y termodinámica

Definición: [ de topología de orden ]

p = min{a,b} <==> ( p [< a & p [< b )

p = max{a,b} <==> ( p >] a || p >] b )

Teorema:

min{max{a,b},w} = max{min{a,w},min{b,w}}

max{min{a,b},w} = min{max{a,w},max{b,w}}

Demostración:

Sea min{max{a,b},w} = p ==>

p = max{a,b} & p = w

( p >] a || p >] b ) & p >] w

( p >] a & p >] w ) || ( p >] b & p >] w )

p >] min{a,w} || p >] min{b,w}

p = max{min{a,w},min{b,w}}

Sea max{min{a,b},w} = p ==>

p = min{a,b} || p = w

( p [< a & p [< b ) || p [< w

( p [< a || p [< w ) & ( p [< b || p [< w )

p [< max{a,w} & p [< max{b,w}

p = min{max{a,w},max{b,w}}

Definición: [ de topología de orden estricto ]

p = sup{a,b} <==> ( p > a || p > b )

p = inf{a,b} <==> ( p < a & p < b )

Teorema:

inf{sup{a,b},w} = sup{inf{a,w},inf{b,w}}

sup{inf{a,b},w} = inf{sup{a,w},sup{b,w}}

Demostración:

Sea inf{sup{a,b},w} = p+(-s) ==>

p+(-s) = sup{a,b} & p+(-s) = w

( p > a || p > b ) & p > w

( p > a & p > w ) || ( p > b & p > w )

p+(-s) > inf{a,w} || p+(-s) > inf{b,w}

p+(-s) = sup{inf{a,w},inf{b,w}}

Sea sup{inf{a,b},w} = p+s ==>

p+s = inf{a,b} || p+s = w

( p < a & p < b ) || p < w

( p < a || p < w ) & ( p < b || p < w )

p+s < sup{a,w} & p+s < sup{b,w}

p+s = inf{sup{a,w},sup{b,w}}

Teorema:

Sea f(x) = x+k ==> f(x) es un morfismo topológico estricto.

Demostración:

f(sup{x,y}) = sup{x,y}+k = sup{x+k,y+k} = sup{f(x),f(y)}

(x+s)+k = (x+k)+s || (y+s)+k = (y+k)+s

f(inf{x,y}) = inf{x,y}+k = inf{x+k,y+k} = inf{f(x),f(y)}

(x+(-s))+k = (x+k)+(-s) || (y+(-s))+k = (y+k)+(-s)

Teorema:

[An][ n·oo^{oo} = oo^{oo} ]

Demostración:

oo^{oo} [< n·oo^{oo} [< oo·oo^{oo} = oo^{oo+1} = oo^{oo}

Teorema:

oo^{oo^{n}} es un cardinal inaccesible irregular.

Demostración:

#( oo^{oo^{n}} ) = oo^{oo}

cof(oo^{oo^{n}}) = oo^{oo^{oo}}

Definición: [ de límite de cardinal inaccesible ]

Sea b_{k} = oo^{oo^{k}} ==>

lim[n = oo][ a_{n} ] = b_{k} <==> ...

... [As][ s > b_{k+(-1)} ==> | ( a_{n}/b_{k} )+(-1)·b_{k+(-1)} | < s ]

Teorema:

Sea k = 1 ==> lim[n = oo][ n^{n^{k}} ] = oo^{oo^{k}}

Demostración:

lim[n = oo][ | ( n^{n^{k}}/oo^{oo^{k}} )+(-1)·oo^{oo^{k+(-1)}} | ] = | oo+(-oo) | = 1 < oo < s

Teorema:

Sea k = 2 ==> lim[n = oo][ n^{n^{k}} ] = oo^{oo^{k}}

Demostración:

lim[n = oo][ | ( n^{n^{k}}/oo^{oo^{k}} )+(-1)·oo^{oo^{k+(-1)}} | ] = oo^{oo+(-1)} = oo^{oo} < s

Teorema:

Sea k >] 3 ==> lim[n = oo][ n^{n^{k}} ] = oo^{oo^{k}}

Demostración:

lim[n = oo][ | ( n^{n^{k}}/oo^{oo^{k}} )+(-1)·oo^{oo^{k+(-1)}} | ] = ...

... oo^{oo^{k+(-1)}+(-1)} = oo^{oo^{k+(-1)}} < s

Teorema:

lim[n = oo][ n! ] = oo^{oo}

Demostración:

[ MP por Stolz ] ==> lim[n = oo][ ( n!/n^{n} )^{(1/n)} ] = e

lim[n = oo][ ( n!/n^{n} ) ] = e^{oo} = oo^{[e]+(-1)} = oo

lim[n = oo][ | ( n!/n^{n} )+(-oo) | ] = | oo+(-oo) | = 1 < oo < s

Definición:

f(x^{2}) = sum[n = 0]-[oo][ [ 2n // n ]·x^{2n} ]

Arte: [ de serie de Laurent ]

[Ex][ f(x^{2}) = 1+sum[n = 1]-[oo][ (-1)^{n}·( (2n)!/(n·n!) )·x^{2n}·e^{nx^{2}} ] ]

Exposición:

x = 0

lim[n = oo][ (2n)! ] = (2·oo)^{2·oo}

f(x) = 1+sum[n = 1]-[oo][ (-1)^{n}·(n+(-1))!·( (2n)!/(n!·n!) )·x^{n}·e^{nx} ]

Arte: [ de Falsus Algebratorum ]

Sea 0 [< x [< 1 ==>

[Ex][ f(x^{2}) = ( 1/(1+(-1)·x^{2}) ) ]

Exposición:

x = 0

[ 2n // n ] = (2n)!·(1/n!)·( 1/(2n+(-n))! ) = (n+n)!·(1/n!)·(1/n!) = ...

... (n+n)!·(n!/n!) = (n+n)! = (n+(-n))! = 0! = 1

Anexo:

Las fracciones continuas de Rogers-Ramanujan,

son destructores que atraviesan estructuras y destruyen por dentro.

Los destructores de función Z(s) de Riemman,

son para las fuerzas eléctricas y gravitatorias.

Teorema:

[ {a},{b} ] = [ 2 // 1 ]

Teorema:

[ {a,b},{b,c},{c,d},{d,a},{a,c},{b,d} ] = [ 4 // 2 ]

Teorema:

[ {a,b,c},{b,c,d},{c,d,e},{d,e,f},{e,f,a},{f,a,b},...

... {a,c,e},{b,d,f},...

... {a,b,d},{b,c,e},{c,d,f},{d,e,a},{e,f,b},{f,a,c},...

... {a,b,e},{b,c,f},{c,d,a},{d,e,b},{e,f,c},{f,a,d} ] = [ 6 // 3 ]

Teorema:

Si [ 2 // 1 ] [<< [ 4 // 2 ] ==> ...

... [EA][ A = [ 2 // 1 ] [& || &] [ 4 // 2 ] & A [<< [ 4 // 2 ] ]

... [EB][ B = A [& || &] [ 6 // 3 ] & B [<< [ 6 // 3 ] ]

Demostración:

Si [ {a},{b} ] [<< [ {a,b},{b,c},{c,d},{d,a},{a,c},{b,d} ] ==> ...

... [ {a},{b} ] [& || &] [ 4 // 2 ] = [ {a,b},{b,d},{d,a},{a,c},{b,c} ]

... [ {a,b},{b,d},{d,a},{a,c},{b,c} ] [& || &] [ 6 // 3 ] = ...

... [ {a,b,c},{b,c,d},{c,d,a},{a,b,d}...

... {a,c,e},{f,a,c},...

... {b,c,e},{b,c,f},...

... {a,b,e},{f,a,b},...

... {d,e,b},{b,d,f}...

... {d,e,a},{f,a,d} ]

Definición:

f(x^{2}) = sum[n = 1]-[oo][ [ 2n // 2n+(-1) ]·x^{2n+(-2)} ]

Arte: [ de serie de Laurent ]

[Ex][ f(x^{2}) = sum[n = 1]-[oo][ (-1)^{n+(-1)}·2·(n+(-1))!·x^{2n+(-2)}·e^{(n+(-1))·x^{2}} ] ]

Exposición:

x = 0

lim[n = oo][ n! ] = oo^{oo}

f(x) = sum[n = 1]-[oo][ (-1)^{n}·(n+(-1))!·2n·x^{n}·e^{nx} ]

Arte: [ de Falsus Algebratorum ]

f(1) = 1

Exposición:

[ 2n // 2n+(-1) ] = 2n = n+n = n+(-n) = 0

Teorema:

Si [ 2 // 1 ] [<< [ 4 // 3 ] ==> ...

... [EA][ A = [ 2 // 2 ] [& || &] [ 4 // 3 ] & A [<< [ 4 // 3 ] ]

... [EB][ B = A [& || &] [ 6 // 5 ] & B [<< [ 6 // 5 ] ]

Demostración:

Si [ {a},{b} ] [<< [ {a,b,c},{b,c,d},{c,d,a},{d,a,b} ] ==> ...

... [ {a,b} ] [& || &] [ 4 // 2 ] = [ {a,b,c},{a,b,d} ]

... [ {a,b,c},{a,b,d} ] [& || &] [ 6 // 5 ] = [ {a,b,c,d,e},{d,e,f,a,b},{e,f,a,b,c},{f,a,b,c,d} ]

Ley:

Conocer a la gente y creer que todos son,

implica el sufrimiento de un rezo al Mal

y entonces también implica el sufrimiento del que reza,

por igualdad del Lucasentismo.

No conocer a la gente o creer que todo-algunos no son,

quizás implica el sufrimiento de un rezo al Mal

pero no implica el sufrimiento del que reza,

por des-igualdad del Lucasentismo.

Deducción: [ por Hobbes-Rousseau ]

Creer que todos son ==> Rezo al Mal

Falsedad ==> Sufrimiento

Creer que todo-algunos no son ==> No rezo al Mal

Verdad ==> Felicidad

Deducción: [ Por Lucasentismo ]

Conocer ==> Confianza ==> Libertad ==> Igualdad

Des-Conocer ==> Des-Confianza ==> Poder y Esclavitud ==> Des-Igualdad

Ley:

Los científicos stronikianos,

son buena gente,

y Dios sigue la Ley con ellos.

Los científicos no stronikianos,

son mala gente,

y Dios no sigue la Ley con ellos.

Deducción:

Se tiene opciones de victoria en la demostración ==> ...

... Confianza ==> Libertad ==> Igualdad ==> Amor

... Luz ==> Se ve que sus obras están hechas como dios quiere

No se tiene opciones de victoria en la demostración ==> ...

... Des-Confianza ==> Poder y esclavitud ==> Des-Igualdad ==> Odior ==> ...

... Tinieblas ==> No se ve que sus obras están hechas como dios quiere

Anexo:

Está demostrado experimentalmente,

no tengo título de matemático y TV3 se salta la Ley con el mapa del tiempo.

Principio: [ de modelo lineal de Lerer-Garriga ]

Sea ( x un estado psicológico & y el estado psicológico dual a x ) ==>

Sean a & b unidades de tiempo ==>

Si f(x,y) = ax+by ==> Ker(f) = { k·< (1/a),(-1)·(1/b) > }

Corrientes en el cerebro de tiempo real:

q(t) = qe^{(1/a)·it}

p(t) = pe^{(-1)·(1/b)·it}

Corrientes en el cerebro de tiempo imaginario:

q(t) = qe^{(-1)·(1/a)·t}

p(t) = pe^{(1/b)·t}

Ley: [ de síndrome psicológico resonante de tiempo real ]

L·d_{tt}^{2}[q]+C·q(t) = We^{(1/a)·it}

q(t) = ( 1/( (-1)·L·(1/a)^{2}+C ) )·We^{(1/a)·it}

L·d_{tt}^{2}[p]+C·p(t) = We^{(-1)·(1/b)·it}

p(t) = ( 1/( (-1)·L·(1/b)^{2}+C ) )·We^{(-1)·(1/b)·it}

Ley: [ de síndrome psicológico resonante de tiempo imaginario ]

L·d_{tt}^{2}[q]+(-C)·q(t) = We^{(-1)·(1/a)·t}

q(t) = ( 1/( L·(1/a)^{2}+(-C) ) )·We^{(-1)·(1/a)·t}

L·d_{tt}^{2}[p]+(-C)·p(t) = We^{(1/b)·t}

p(t) = ( 1/( L·(1/b)^{2}+(-C) ) )·We^{(1/b)·t}

Ley: [ de síndrome psicológico anti-resonante de tiempo real ]

( L·d_{tt}^{2}[q]+C·q(t) )·(1/q(t))^{2} = (1/q)^{2}·We^{(1/a)·it}

q(t) = q^{2}·( (-1)·L·(1/a)^{2}+C )·(1/W)·e^{(-1)·(1/a)·it}

( L·d_{tt}^{2}[p]+C·p(t) )·(1/p(t))^{2} = (1/p)^{2}·We^{(-1)·(1/b)·it}

p(t) = p^{2}·( (-1)·L·(1/b)^{2}+C )·(1/W)·e^{(1/b)·it}

Ley: [ de síndrome psicológico anti-resonante de tiempo imaginario ]

( L·d_{tt}^{2}[q]+(-C)·q(t) )·(1/q(t))^{2} = (1/q)^{2}·We^{(-1)·(1/a)·t}

q(t) = q^{2}·( L·(1/a)^{2}+(-C) )·(1/W)·e^{(1/a)·t}

( L·d_{tt}^{2}[p]+(-C)·p(t) )·(1/p(t))^{2} = (1/p)^{2}·We^{(1/b)·t}

p(t) = p^{2}·( L·(1/b)^{2}+(-C) )·(1/W)·e^{(-1)·(1/b)·t}

Ley: [ de síndrome psicológico resonante con fase de tiempo real ]

L·d_{tt}^{2}[q]+C·q(t) = W·( e^{(1/a)·it}+sin((1/a)·t) )

q(t) = ( 1/( (-1)·L·(1/a)^{2}+C ) )·W·( e^{(1/a)·it}+sin((1/a)·t) )

L·d_{tt}^{2}[p]+C·p(t) = W·( e^{(-1)·(1/b)·it}+cos((-1)·(1/b)·t) )

p(t) = ( 1/( (-1)·L·(1/b)^{2}+C ) )·W·( e^{(-1)·(1/b)·it}+cos((-1)·(1/b)·t) )

Ley: [ de síndrome psicológico resonante con fase de tiempo imaginario ]

L·d_{tt}^{2}[q]+(-C)·q(t) = W·( e^{(-1)·(1/a)·t}+sinh((-1)·(1/a)·t) )

q(t) = ( 1/( L·(1/a)^{2}+(-C) ) )·W·( e^{(-1)·(1/a)·t}+sinh((-1)·(1/a)·t) )

L·d_{tt}^{2}[p]+(-C)·p(t) = W·( e^{(1/b)·t}+cosh((1/b)·t) )

p(t) = ( 1/( L·(1/b)^{2}+(-C) ) )·W·( e^{(1/b)·t}+cosh((1/b)·t) )

Principio: [ de modelo lineal de Lerer-Garriga ]

Sea ( (x un estado psicológico & y el estado psicológico dual a x ) & ...

... z un estado psicológico dual a ( x & y ) ) ==>

Sean a & b & c unidades de tiempo ==>

Si f(x,y,z) = ax+by+cz ==> Ker(f) = { k·< (1/(2a)),(1/(2b)),(-1)·(1/c) > }

Corrientes en el cerebro de tiempo real:

q(t) = q·( e^{(1/(2a))·it}+sin((-1)·(1/c)·t) )

p(t) = p·( e^{(1/(2b))·it}+cos((-1)·(1/c)·t) )

Corrientes en el cerebro de tiempo imaginario:

q(t) = q·( e^{(-1)·(1/(2a))·t}+sinh((1/c)·t) )

p(t) = p·( e^{(-1)·(1/(2b))·t}+cosh((1/c)·t) )

Ley: [ de síndrome psicológico resonante con fase de tiempo real ]

L·d_{tt}^{2}[q]+C·q(t) = W·( e^{(1/(2a))·it}+sin((-1)·(1/c)·t) )

q(t) = ...

... ( 1/( (-1)·L·(1/(2a))^{2}+C ) )·W·e^{(1/(2a))·it}+( 1/( (-1)·L·(1/c)^{2}+C ) )·W·sin((-1)·(1/c)·t) )

L·d_{tt}^{2}[p]+C·p(t) = W·( e^{(1/(2b))·it}+cos((-1)·(1/c)·t) )

p(t) = ...

... ( 1/( (-1)·L·(1/(2b))^{2}+C ) )·W·e^{(1/(2b))·it}+( 1/( (-1)·L·(1/c)^{2}+C ) )·W·cos((-1)·(1/c)·t) )

Ley: [ de síndrome psicológico resonante con fase de tiempo imaginario ]

L·d_{tt}^{2}[q]+(-C)·q(t) = W·( e^{(-1)·(1/(2a))·t}+sinh((1/c)·t) )

q(t) = ...

... ( 1/( L·(1/(2a))^{2}+(-C) ) )·W·e^{(-1)·(1/(2a))·t}+( 1/( L·(1/c)^{2}+(-C) ) )·W·sinh((1/c)·t) )

L·d_{tt}^{2}[p]+(-C)·p(t) = W·( e^{(-1)·(1/(2b))·t}+cosh((1/c)·t) )

p(t) = ...

... ( 1/( L·(1/(2b))^{2}+(-C) ) )·W·e^{(-1)·(1/(2b))·t}+( 1/( L·(1/c)^{2}+(-C) ) )·W·cosh((1/c)·t) )

Principio: [ psicológico musical de Lerer-Garriga ]

Sea ( x un estado psicológico & y el estado psicológico dual a x ) ==>

Sean a & b unidades de tiempo ==>

Si f(x,y) = ax+by ==> Ker(f) = { k·< (1/a),(-1)·(1/b) > }

Terapia musical:

[...]...(n)...[...] = f(a)

[...]...(n)...[...] = g(b)

Ley:

Sea x estar despierto & y estar dormido ==>

Si ( a = 19 & b = 5 ) ==> 

[01][05][08][05] = 19

[07][11][14][11] = 43

Falsus Algebratorum:

43 = 24+19 = 24+(-19) = 5

Ley:

Sea x estar despierto & y estar dormido ==>

Si ( a = 17 & b = 7 ) ==> 

[01][04][08][04] = 17

[07][10][14][10] = 41

Falsus Algebratorum:

41 = 24+17 = 24+(-17) = 7

Ley:

Sea x estar en ayuno & y pasar hambre ==>

Si ( a = 13 & b = 5 ) ==>

[03][07][03] = 13

[09][13][09] = 31

Falsus Algebratorum:

31 = 18+13 = 18+(-13) = 5

Ley:

Sea x estar en ayuno & y pasar hambre ==>

Si ( a = 11 & b = 7 ) ==> 

[02][07][02] = 11

[08][13][08] = 29

Falsus Algebratorum:

29 = 18+11 = 18+(-11) = 7

Ley:

Sea x estar solo & y estar acompañado ==>

Si ( a = 23 & b = 7 ) ==> 

[04][...][04][...][04][07][04][...] = 23 = 20+3 = 2·2·5+3

[10][...][10][...][10][13][10][...] = 53 = 50+3 = 2·5·5+3

Falsus Algebratorum:

53 = 30+23 = 30+(-23) = 7

Ley:

Sea x estar solo & y estar acompañado ==>

Si ( a = 17 & b = 13 ) ==> 

[03][...][03][...][03][05][03][...] = 17 = 12+3 = 4·3+3

[09][...][09][...][09][11][09][...] = 47 = 44+3 = 4·11+3

Falsus Algebratorum:

47 = 30+17 = 30+(-17) = 13

Principio: [ polinómico de drogadicción de Lerer-Garriga ]

Sea ( x un estado de drogadicción & b las experiencias vividas drogado ) ==>

Sea a unidad de tiempo ==>

Si f( P(x) ) = P(x)+ax+(-b) ==> Ker(f) = { (-1)·ax+b }

Sea ( x^{2} un doble estado de drogadicción & b las experiencias vividas drogado ) ==>

Sea a unidad de tiempo ==>

Si f( P(x) ) = P(x)+acx^{2}+(-b) ==> Ker(f) = { (-1)·ax+b }

Corriente en el cerebro de drogadicción:

q(t) = qe^{(1/a)·t}

p(t) = pe^{(1/a)·it}

Corriente en el cerebro de anti-drogadicción:

P(t) = pe^{(-1)·(1/a)·t}

Q(t) = qe^{(-1)·(1/a)·it}

Ley: [ de invariante Gauge ]

q(t)·P(t) = p(t)·Q(t)

Principio: [ polinómico de drogadicción de Lerer-Garriga ]

Sea ( x^{2} un doble estado de drogadicción & b las experiencias vividas drogado ) ==>

Sea a unidad de tiempo ==>

Si f( P(x) ) = P(x)+acx^{2}+(-b) ==> Ker(f) = { (-1)·acx^{2}+b }

Corriente en el cerebro de drogadicción:

q(t) = qe^{(1/(ac))^{(1/2)}·t}

p(t) = pe^{(1/(ac))^{(1/2)}·it}

Corriente en el cerebro de anti-drogadicción:

P(t) = pe^{(-1)·(1/(ac))^{(1/2)}·t}

Q(t) = qe^{(-1)·(1/(ac))^{(1/2)}·it}

Ley: [ de invariante Gauge ]

q(t)·P(t) = p(t)·Q(t)

Ley: [ de Alcohólicos Anónimos ]

Sea f( P(x) ) = P(x)+ax+(-b) ==>

Sea g( P(x) ) = P(x)+acx^{2}+(-b) ==>

Sea a = 0 ==>

Si h( P(x) ) = P(x)+(-b) ==> Ker(f) = {b}

No habla el terapeuta.

Ley: [ de Parches de Nicotina Anti-Tabaco ]

Sea f( P(x) ) = P(x)+ax+(-b) ==>

Sea (-b) = 0 ==>

Si h( P(x) ) = P(x)+ax ==> Ker(f) = {(-1)·ax}

Ley: [ de Hierba-Luisa Anti-Marihuana ]

Sea f( P(x) ) = P(x)+acx^{2}+(-b) ==>

Sea (-b) = 0 ==>

Si h( P(x) ) = P(x)+acx^{2} ==> Ker(f) = {(-1)·acx^{2}}

Psico-Neurología o Psiquiatría:

Primero:

Cálculo diferencial.

Tablas de la Verdad y Álgebra lineal.

Segundo:

Cálculo integral.

Psico-neurología de negación esquizofrénica.

Tercero:

Psico-neurología de destructor de doble mandamiento.

Psico-neurología resonante y anti-resonante.

Cuarto:

Psico-neurología de drogadicción.

Psico-neurología musical.

Medicina:

Primero:

Cálculo diferencial.

Química.

Segundo:

Cálculo integral.

Física Termodinámica y Cabal.

Tercero:

Teoría genética de virus.

Teoría genética de bacterias.

Cuarto:

Quimioterapia

Espectroscopia y Láser Quirúrgico

Quinto:

Óptica de lentes.

Oftalmología de vista y oída.

Sexto:

Traumatología.

Oncología.

Teorema:

x(t) [o(t)o] y(t) = f(x(t)) & f(t) = t^{n} ==>

Dual[A] = { < t^{(1/n)}, ( t /o(t)o/ t^{(1/n)} ) > }

Teorema:

x(t) [o(t)o] y(t) = f(x(t)) & f(t) = e^{nt} ==>

Dual[A] = { < (1/n)·ln(t), ( t /o(t)o/ (1/n)·ln(t) ) > }

Indulgencia:

Teorema:

Quitzare-tur sere-tur perdunatered omnia pecatorum,

després-ne-tur de pagare-tur condenaziorum

y aleshorum tambene-tur espiritus blafemium 

sere-tur perdunatered aduquene-torum,

Omnia y existere-tur not sere-tur contradicziorum.

Demostraziorum:

Sere-tur perdunatered omnia pecatorum,

després-ne-tur de pagare-tur condenaziorum

prum espiritus blafemium

not sere-tur perdunatered aduquene-torum.

Omnia y not existere-tur sere-tur contradicziorum.

Láser Quirúrgico:

Ley:

T·d_{t}[q] = ((act)^{n}+1)·q(t)·(1/p)·hf·cos(act)

q(t) = qe^{(1/T)·(1/(n+1))·( (act)^{n+1}+act )·(1/(pac))·hf [o(act)o] sin(act)}

Sea x(t) = ct ==>

q(t) = qe^{(1/T)·( (1/(n+1))·(ax)^{n+1}+ax )·(1/(pac))·hf [o(ax)o] sin(ax)}

Ley:

T·d_{t}[q] = ((act)^{n}+1)·q(t)·(1/p)·hf·(-1)·sin(act)

q(t) = qe^{(1/T)·(1/(n+1))·( (act)^{n+1}+act )·(1/(pac))·hf [o(act)o] cos(act)}

Sea x(t) = ct ==>

q(t) = qe^{(1/T)·( (1/(n+1))·(ax)^{n+1}+ax )·(1/(pac))·hf [o(ax)o] cos(ax)}

Ley:

No se puede seguir televisión,

porque la imagen es de un oscilador,

de imagen de código genético.

No se puede seguir la radio,

porque el sonido es de un oscilador,

de sonido de código genético.

Teorema:

lim[x = 1][ ( 1/(x^{q}+(-1)) )·( x^{p}+(-1) ) ] = (p/q)

Demostración:

x^{k}+(-1) = (x+(-1))·( x^{k+(-1)}+...(k)...+1 )

1+...(p)...+1 = p

1+...(q)...+1 = q

Arte:

lim[x = 1][ ( 1/(x^{q}+(-1)) )·( x^{p}+(-1) ) ] != (q/p)

Exposición:

(p/q) = pq = (q/p)

Teorema:

lim[x = 1][ ( 1/(x+(-1)) )^{2n}·( (x+(-1))·...(2n)...·(x^{2n}+(-1)) ) ] = (2n)!

Demostración:

x^{k}+(-1) = (x+(-1))·( x^{k+(-1)}+...(k)...+1 )

1·...·(1+...(n)...+1)·(1+...(n+1)...+1)·...·(1+...(2n)...+1) = 1·...n·(n+1)·...·2n = (2n)!

Arte:

lim[x = 1][ ( 1/(x+(-1)) )^{2n}·( (x+(-1))·...(2n)...·(x^{2n}+(-1)) ) ] != 1

Exposición:

(2n)! = (n+n)! = (n+(-n))! = 0! = 1

Teorema:

lim[x = 1][ (1/(x+(-1)))·( (x+...(n)...+x^{n})+(-n) ) ] = (1/2)·n·(n+1)

Demostración:

x^{k}+(-1) = (x+(-1))·( x^{k+(-1)}+...(k)...+1 )

1+...+(1+...(n)...+1) = 1+...(n)...+n = (1/2)·n·(n+1)

Arte:

lim[x = 1][ (1/(x+(-1)))·( (x+...(n)...+x^{n})+(-n) ) ] != 2n·(n+(-1))

Exposición:

(1/2)·n·(n+1) = 2n·(n+1) = 2n·(n+(-1))

Ley:

Sistema-A de falso testimonio:

Es toto-hoimbre que se conoce.

Jûan Garriga no es maricón,

porque es.

Los transexuales son.

Sistema-B de falso testimonio:

No es ninguien que se conoce.

Jûan Garriga es maricón,

porque no es.

Los transexuales no son.

Anexo:

No es tan fácil decir,

que Jûan Garriga es maricón,

sin condenación.

Ley:

Sistema-A de falso testimonio:

No es todo-alguien que se conoce.

Jûan Garriga viola mentalmente a una mujer,

que es.

Jûan Garriga no viola mentalmente a mujeres,

que no son.

Sistema-B de falso testimonio:

Es alguien que se conoce.

Jûan Garriga no viola mentalmente a una mujer,

que es.

Jûan Garriga viola mentalmente a mujeres,

que no son.

Anexo:

Estos sistemas son de falso testimonio,

y no seguir la Lógica es condenación,

como es inventar-se las matemáticas.

Creyendo que es toto-hoimbre,

no se puede decir que Jûan Garriga viola a mujeres,

porque no hay negación de falso testimonio y no es destructor.

Creed-vos que soy maricón y que no soy,

y marchad-vos a vuestro mundo,

que no vais a vencer a Dios el Creador.

Ley:

Sistema-A de falso testimonio:

El padre de un señor es un caballo,

teniendo el señor esa o aquella picha.

El padre de un caballo es un caballo,

teniendo el caballo esa o aquella picha.

Sistema-B de falso testimonio:

El padre de una señora es un caballo,

teniendo la señora ese o aquel chocho.

El Padre de una yegua es un caballo,

teniendo la yegua ese o aquel chocho.

Ley:

Sea x(t) = v·cos(w)·t & y(t) = (-1)·g·(1/2)·t^{2}+v·sin(w)·t ==>

( x(t_{k}) = d & y(t_{k}) = h ) <==> ...

... w = arc-tan( (1/(gd))·v^{2}·( 1+( 1+(-1)·(1/v)^{2}·g·( 2h+(d/v)^{2}·g ) )^{(1/2)} ) )

2h+(d/v)^{2}·g = (1/g)·v^{2} <==> w = arc-tan( (1/(gd))·v^{2} )

Deducción:

h = (-1)·g·(1/2)·(d/v)^{2}·( 1+( tan(w) )^{2} )+tan(w)·d

Ley:

Sea x(t) = v·cos(w)·t & y(t) = (-1)·g·(1/2)·t^{2}+v·sin(w)·t ==>

y(t) = 0 <==> x(w) = (-1)·(1/2)·(1/g)·v^{2}·sin(w)·cos(w)

x(w) es máximo <==> w = (pi/4)

Historia:

Melkor fue venciendo,

hasta que llegó al máximo de su condenación.

Tulkas solo lo encadenó,

y las cadenas lo llevaron a una zona muerta vacía,

porque Melkor se creía que la verdad implicaba el sufrimiento,

y esto lo condenó a una zona muerta a estar solo.

Si no se paga la condenación te extingues como Melkor.

Teorema:

Sea d_{x}[F(x)] = f(x) ==>

Si lim[x = oo][ F(x) ] = c ==> lim[y = oo][ int[x = a]-[b][ f(x+y) ]d[x] ] = 0c

Demostración:

lim[y = oo][ F(b+y)+(-1)·F(a+y) ] = F(b+oo)+(-1)·F(a+oo) = F(oo)+(-1)·F(oo) = c+(-c) = 0c

Teorema:

Sea d_{x}[F(x)] = f(x) ==>

Si lim[x = (-oo)][ F(x) ] = c ==> lim[y = (-oo)][ int[x = a]-[b][ f(x+y) ]d[x] ] = 0c

Demostración:

lim[y = (-oo)][ F(b+y)+(-1)·F(a+y) ] = F(b+(-oo))+(-1)·F(a+(-oo)) = F(-oo)+(-1)·F(-oo) = c+(-c) = 0c

Arte:

Sea d_{x}[F(x)] = f(x) ==>

Si F(0) = 2c ==> lim[x = 1][ int[ f(2x) ]d[x] ] != c

Exposición:

int[ f(2x) ]d[x] = int[ (1/2)·f(2x)·2 ]d[x] = (1/2)·int[ f(2x)·2 ]d[x] = (1/2)·F(2x)

lim[x = 1][ (1/2)·F(2x) ] = lim[x = 1][ (1/2)·F(x+x) ] = lim[x = 1][ (1/2)·F(x+(-x)) ] = ...

... (1/2)·F(1+(-1)) = (1/2)·F(0) = (1/2)·2c = c

Arte:

Sea d_{x}[F(x)] = f(x) ==>

Si F(1) = c ==> lim[x = 1][ int[ f((2n)!·x) ]d[x] ] != c

Exposición:

int[ f((2n)!·x) ]d[x] = int[ (1/(2n)!)·f((2n)!·x)·(2n)! ]d[x] = ...

... (1/(2n)!)·int[ f((2n)!·x)·(2n)! ]d[x] = (1/(2n)!)·F((2n)!·x)

lim[x = 1][ (1/(2n)!)·F((2n)!·x) ] = lim[x = 1][ (1/(n+n)!)·F((n+n)!·x) ] = ...

... lim[x = 1][ ( 1/(n+(-n))! )·F((n+(-n))!·x) ] = lim[x = 1][ (1/0!)·F(0!·x) ] = ...

... lim[x = 1][ F(x) ] = F(1) = c

Ley:

Sea VP = kT ==>

d_{P}[T(P,V)] = (T/P)

d_{V}[T(P,V)] = (T/V)

d_{P}[T(P,V)]·p = qR <==> p = qR·(P/T)

d_{V}[T(P,V)]·v = qR <==> v = qR·(V/T)

Ley:

Sea aP^{2}+bV^{2} = kT ==>

d_{PP}^{2}[T(P,V)] = (1/P)·d_{P}[T(P,V)]

d_{VV}^{2}[T(P,V)] = (1/V)·d_{V}[T(P,V)]

d_{PP}^{2}[T(P,V)]·p^{2} = qR <==> p = ( qRP·( 1/d_{P}[T(P,V)] ) )^{(1/2)}

d_{VV}^{2}[T(P,V)]·v^{2} = qR <==> v = ( qRV·( 1/d_{V}[T(P,V)] ) )^{(1/2)}

Ley:

Sea Ue^{aP+bV} = kT ==>

d_{P...P}^{n}[T(P,V)] = Ta^{n}

d_{V...V}^{n}[T(P,V)] = Tb^{n}

d_{P...P}^{n}[T(P,V)]·p^{n} = qR <==> p = ( qR·(1/a)^{n}·(1/T) )^{(1/n)}

d_{V...V}^{n}[T(P,V)]·v^{n} = qR <==> v = ( qR·(1/b)^{n}·(1/T) )^{(1/n)}

Deducción:

d_{P...P}^{n+1}[T(P,V)] = d_{P}[Ta^{n}] = d_{P}[T]·a^{n} = (Ta)·a^{n} = Ta^{n+1}

d_{V...V}^{n+1}[T(P,V)] = d_{V}[Tb^{n}] = d_{V}[T]·b^{n} = (Tb)·b^{n} = Tb^{n+1}

Ley:

Sea Ue^{aP+bV} = kT ==>

d_{T...T}^{n}[P(T,V)] = (-1)^{n+1}·(n+(-1))!·(1/T)^{n}·(1/a)

d_{T...T}^{n}[V(P,T)] = (-1)^{n+1}·(n+(-1))!·(1/T)^{n}·(1/b)

d_{T...T}^{n}[P(T,V)]·s^{n} = ( F/(4pi·R^{2}) ) <==> ...

... s = ( ( F/(4pi·R^{2}) )·a·(-1)^{(-n)+(-1)}·( 1/(n+(-1))! )·T^{n} )^{(1/n)}

d_{T...T}^{n}[V(P,T)]·s^{n} = (4/3)·pi·R^{3} <==> 

... s = ( ( (4/3)·pi·R^{3} )·b·(-1)^{(-n)+(-1)}·( 1/(n+(-1))! )·T^{n} )^{(1/n)}

Deducción:

d_{T...T}^{n+1}[P(T,V)] = d_{T}[(-1)^{n+1}·(n+(-1))!·(1/T)^{n}·(1/a)] = ...

... (-1)^{n+1}·(n+(-1))!·d_{P}[(1/T)^{n}]·(1/a) = (-1)^{(n+1)+1}·n!·(1/T)^{n+1}·(1/a)

arte-matemático y medicina y falsus-infinitorum y falsus-algebratorum y números-irracionales y computación

Arte:

[En][ ( n+(1/(n+1)) ) [< (3/2) ]

Exposición:

n = 1

f(n) = 0

1 = ( f(n)+(1/(f(n)+1)) ) = ( n+(1/(n+1)) ) < (3/2)

Arte:

[En][ ( 2n+(1/(2n+1)) ) [< (7/3) ]

Traumatología vertebral:

Principio:

Amisnostrofia resistiva:

[ER][ R(t) >] 1 & W = R(t)·d_{t}[q] ]

Principio:

Aminostrofia condensativa:

[EC][ 0 [< C(t) [< 1 & W = C(t)·q(t) ]

Traumatología sanguínea:

Principio:

Trompo de colesterol:

Sea K es cabal sanguíneo ==>

[ES][ S(t) >] 1 & K = S(t)·d_{t}[x] ]

Principio:

Trompo de obstrucción de arteria:

Sea K es cabal sanguíneo ==>

[ES][ S(t) >] 1 & K = S(t)·ux ]

Oncología de transporte de intestino:

Principio:

Flojura de estómago:

Sea d_{xyz}^{3}[m(x,y,z,t)] la densidad del cagado ==>

[EM][ (M(t)/K) = int[ d_{xyz}^{3}[m(x,y,z,t)] ]d[t] ]

Principio:

Dureza de riñón:

Sea d_{xyz}^{3}[m(x,y,z,t)] la densidad del pijado ==>

[EM][ (M(t)/K) = int[ d_{xyz}^{3}[m(x,y,z,t)] ]d[t] ]

Ley:

Sea d_{xyz}^{3}[m(x,y,z,t)] = ma^{3} ==>

M(t) = mKa^{3}·t

m(x,y,z,t) = maxayaz

Ley:

Sea t >] (1/u) ==>

Si ( ( Si vt >] 1 ( retortijón ) ==> z(t) = vt ) & d_{xyz}^{3}[m(x,y,z,t)] = ma^{2}·(1/z)·(ut) ) ==>

M(t) = mKa^{2}·(1/v)·(ut)

m(x,y,z,t) = maxay·(ut)

Deducción:

d[z] = v·d[t]

d[x]d[y]d[z] = v·d[x]d[y]d[t]

Ley:

Sea t >] (1/u) ==>

Si ( ( Si vt >] 1 ( retortijón ) ==> z(t) = vt ) & d_{xyz}^{3}[m(x,y,z,t)] = ma^{4}·z·(1/(ut)) ) ==>

M(t) = mKa^{4}·v·(1/u)·t

m(x,y,z,t) = maxay·(av)^{2}·(1/u)·t

Deducción:

d[z] = v·d[t]

d[x]d[y]d[z] = v·d[x]d[y]d[t]

Ley:

Sea t >] (1/u) ==>

Si ( ( Si vt >] 1 ==> z(t) = vt ) & d_{xyz}^{3}[m(x,y,z,t)] = ma^{2}·(1/z)·(1/2)·(1+(ut)) ) ==>

M(t) = mKa^{2}·(1/(2v))·( ln(ut)+ut )

m(x,y,z,t) = maxay·(1/2)·( ln(ut)+ut )

Ley:

Sea t >] (1/u) ==>

Si ( ( Si vt >] 1 ==> z(t) = vt ) & d_{xyz}^{3}[m(x,y,z,t)] = ma^{4}·2z·( 1/(1+(ut)) ) ) ==>

M(t) = mKa^{4}·((2v)/u)·( ln(1+ut) [o(ut)o] (1/2)·(ut)^{2} )

m(x,y,z,t) = maxay·2·(av)^{2}·(1/u)^{2}·( ln(1+ut) [o(ut)o] (1/2)·(ut)^{2} )

Teoría del Falsus Infinitorum:

Axioma:

Si [Ej][ lim[n = j][ a_{n} ] = a & f(j) = m & g(m) = oo ] ==> ¬( lim[n = g(f(j))][ a_{n} ] = a )

Teoría del Falsus Algebratorum:

Axioma:

Si [Aj][ lim[n = j][ a_{n} ] != a & f(j) = m & g(m) = oo ] ==> ¬( lim[n = g(f(j))][ a_{n} ] = a )

Euler Falsus Infinitorum:

Arte:

sum[k = 0]-[oo][ ( 1/(k+1) )^{m} ] = (1/2)^{m}

Exposición:

lim[n = 1][ sum[k = 0]-[n][ ( 1/(k+1) )^{m} ]+(-1)·(1/n) ] = 1+(1/2)^{m}+(-1) = (1/2)^{m}

Arte:

sum[k = 1]-[oo][ ( 1/ln(k+1) )^{m}+(-1)·(1/k) ] = ( 1/ln(2) )^{m}

Exposición:

lim[n = 1][ sum[k = 1]-[n][ ( 1/ln(k+1) )^{m}+(-1)·(1/k) ]+(1/n) ] = ...

... ( 1/ln(2) )^{m}+(-1)+1 = ( 1/ln(2) )^{m}

Arte:

sum[k = 1]-[oo][ ( ln((1/k)+1) )^{m}+(-1)·(1/k) ] = ( ln(2) )^{m}

Exposición:

lim[n = 1][ sum[k = 1]-[n][ ( ln((1/k)+1) )^{m}+(-1)·(1/k) ]+(1/n) ] = ...

... ( ln(2) )^{m}+(-1)+1 = ( ln(2) )^{m}

Arte:

Prod[k = 1]-[oo][ (1/(2p))^{k}+( 1/(k+1) )^{m} ] = (1/p)+(1/2)^{m+(-1)}

Exposición:

lim[n = 1][ Prod[k = 1]-[n][ (1/(2p))^{k}+( 1/(k+1) )^{m} ]·(1+(1/n)) ] = ...

... ( (1/(2p))+(1/2)^{m} )·2 = (1/p)+(1/2)^{m+(-1)}

Arte:

Prod[k = 0]-[oo][ (1/(4p))^{k}+( 1/(k+1) )^{m} ] = (1/p)+(1/2)^{m+(-2)}

Exposición:

lim[n = 1][ Prod[k = 0]-[n][ (1/(4p))^{k}+( 1/(k+1) )^{m} ]·(1+(1/n)) ] = ...

... ( (1/(4p))+(1/2)^{m} )·4 = (1/p)+(1/2)^{m+(-2)}

Arte:

Prod[k = 1]-[oo][ m·( 1/(k+1) ) ] = m

Exposición:

lim[n = 1][ Prod[k = 1]-[n][ m·( 1/(k+1) ) ]·(1+(1/n)) ] = ( m·(1/2) )·2 = m

Arte:

Prod[k = 0]-[oo][ m·( 1/(k+1) ) ] = m^{2}

Exposición:

lim[n = 1][ Prod[k = 0]-[n][ m·( 1/(k+1) ) ]·(1+(1/n)) ] = m·( m·(1/2) )·2 = m^{2}

Ramanujan Falsus Infinitorum:

Arte:

Sea Z(s) = sum[k = 1]-[oo][ (1/k)^{s} ] ==>

Z(2)·sum[k = 1]-[oo][ 17·(1/k)^{2}+(-11)·(1/k)^{3} ]·(1/13) = pi^{2}

Exposición:

lim[n = 1][ Z(2)·sum[k = 1]-[n][ 17·(1/k)^{2}+(-11)·(1/k)^{3} ]·(1/13)+(12/13)·(1/n)·pi^{2} ] = ...

... Z(2)·(6/13)+(12/13)·pi^{2} = pi^{2}

Arte:

Sea Z(s) = sum[k = 1]-[oo][ (1/k)^{s} ] ==>

Z(2)·sum[k = 1]-[oo][ 47·(1/k)^{2}+(-41)·(1/k)^{3} ]·(1/43) = pi^{2}

Exposición:

lim[n = 1][ Z(2)·sum[k = 1]-[n][ 47·(1/k)^{2}+(-41)·(1/k)^{3} ]·(1/43)+(42/43)·(1/n)·pi^{2} ] = ...

... Z(2)·(6/43)+(42/43)·pi^{2} = pi^{2}

Arte:

Sea Z(s) = sum[k = 1]-[oo][ (1/k)^{s} ] ==>

Z(4)·sum[k = 1]-[oo][ 113·(1/k)^{4}+(-23)·(1/k)^{5} ]·(61/71) = pi^{4}

Exposición:

lim[n = 1][ Z(4)·sum[k = 1]-[n][ 113·(1/k)^{4}+(-23)·(1/k)^{5} ]·(61/71)+(10/71)·(1/n)·pi^{4} ] = ...

... Z(4)·90·(61/71)+(10/71)·pi^{4} = pi^{4}

Arte:

Sea Z(s) = sum[k = 1]-[oo][ (1/k)^{s} ] ==>

Z(4)·sum[k = 1]-[oo][ 101·(1/k)^{4}+(-11)·(1/k)^{5} ]·(43/73) = pi^{4}

Exposición:

lim[n = 1][ Z(4)·sum[k = 1]-[n][ 101·(1/k)^{4}+(-11)·(1/k)^{5} ]·(43/73)+(30/73)·(1/n)·pi^{4} ] = ...

... Z(4)·90·(43/73)+(30/73)·pi^{4} = pi^{4}

Arte:

Sea Z(s) = sum[k = 1]-[oo][ (1/k)^{s} ] ==>

Z(2)·sum[k = 1]-[oo][ ( (4k)!/k! )·( (53k+1)/216^{k} ) ]·(17/71) = pi^{2}

Exposición:

lim[n = 1][ Z(2)·sum[k = 1]-[n][ ( (4k)!/k! )·( (53k+1)/216^{k} ) ]·(17/71)+...

... (54/71)·(1/n)·pi^{2} ] = Z(2)·4!·54·(1/(4·54))·(17/71)+(54/71)·pi^{2} = pi^{2}

Arte:

Sea Z(s) = sum[k = 1]-[oo][ (1/k)^{s} ] ==>

Z(2)·sum[k = 1]-[oo][ ( (4k)!/k! )·( (73k+(-1))/288^{k} ) ]·(19/91) = pi^{2}

Exposición:

lim[n = 1][ Z(2)·sum[k = 1]-[n][ ( (4k)!/k! )·( (73k+(-1))/288^{k} ) ]·(19/91)+...

... (72/91)·(1/n)·pi^{2} ] = Z(2)·4!·72·(1/(4·72))·(19/91)+(72/91)·pi^{2} = pi^{2}

Arte:

Sea Z(s) = sum[k = 1]-[oo][ (1/k)^{s} ] ==>

Z(s) = (9/2)·Z(s)·sum[k = 1]-[oo][ e^{(-3)·pi·(k+(-1))} ]+

... (-2)·Z(s)·sum[k = 1]-[oo][ e^{(-2)·pi·(k+(-1))} ]+...

... (-1)·(1/2)·Z(s)·sum[k = 1]-[oo][ e^{(-pi)·(k+(-1))} ]

Exposición:

lim[n = 1][ (9/2)·Z(s)·sum[k = 1]-[n][ e^{(-3)·pi·(k+(-1))} ]+...

... (-2)·Z(s)·sum[k = 1]-[n][ e^{(-2)·pi·(k+(-1))} ]+...

... (-1)·(1/2)·Z(s)·sum[k = 1]-[n][ e^{(-pi)·(k+(-1))} ]+(-1)·Z(s)·(1/n) ] = Z(s)

Arte:

Sea Z(s) = sum[k = 1]-[oo][ (1/k)^{s} ] ==>

Z(s) = (31/8)·Z(s)·sum[k = 1]-[oo][ e^{(-31)·pi·(k+(-1))} ]+...

... (-2)·Z(s)·sum[k = 1]-[oo][ e^{(-2)·pi·(k+(-1))} ]+...

... (1/8)·Z(s)·sum[k = 1]-[oo][ e^{(-pi)·(k+(-1))} ]

Arte:

( 1+sum[k = 1]-[oo][ k·cos(k+(-1)) ] )^{(-2)}+( 1+sum[k = 1]-[oo][ k·cosh(k+(-1)) ] )^{(-2)} = 1

Exposición:

lim[n = 1][ ( 1+sum[k = 1]-[n][ k·cos(k+(-1)) ] )^{(-2)}+...

... ( 1+sum[k = 1]-[n][ k·cosh(k+(-1)) ] )^{(-2)}+(1/(2n)) ] = 1

Arte:

( 1+sum[k = 1]-[oo][ k·( 1/(k+(-1)) )·sin(k+(-1)) ] )^{(-2)}+...

... ( 1+sum[k = 1]-[oo][ k·( 1/(k+(-1)) )·sinh(k+(-1)) ] )^{(-2)} = 1

Arte:

Frac[k = 0]-[oo][ ( k!/(1+(k+1)!) ) ] = 1

Exposición:

lim[n = 0][ Frac[k = 0]-[n][ ( k!/(1+(k+1)!) ) ]·( 1+(1/(n+1)) ) ] = 1

Arte:

Frac[k = 0]-[oo][ ( (2k+1)/(1+(2k+3)) ) ] = (1/2)

Arte:

Frac[k = 1]-[oo][ ( (2k)/(1+(2k+2)) ) ] = (4/5)

Arte:

ln( Frac[k = 0]-[oo][ ( (e^{(-k)}+(-k))/(1+e^{(-1)·(k+1)}+(-1)·(k+1)) ) ] ) = ln(2)+1

Exposición:

lim[n = 0][ ...

... ln( Frac[k = 0]-[n][ ( (e^{(-k)}+(-k))/(1+e^{(-1)·(k+1)}+(-1)·(k+1)) ) ]·( 1+(1/(n+1)) ) ) ] = ...

.. ln(2e) = ln(2)+ln(e) = ln(2)+1

Arte:

ln( Frac[k = 0]-[oo][ ( (e^{(-k)·pi}+(-k))/(1+e^{(-1)·(k+1)·pi}+(-1)·(k+1)) ) ] ) = ln(2)+pi

Exposición:

lim[n = 0][ ...

... ln( Frac[k = 0]-[n][ ( (e^{(-k)·pi}+(-k))/(1+e^{(-1)·(k+1)·pi}+(-1)·(k+1)) ) ]·( 1+(1/(n+1)) ) ) ] = ...

... ln(2e^{pi}) = ln(2)+ln(e^{pi}) = ln(2)+pi

Arte:

ln( Frac[k = 1]-[oo][ ( (e^{k}+(-k))/(2k+e^{(k+1)}+(-1)·(k+1)) ) ] ) = ln(2)+(-2)+[1:(-1)]

Exposición:

lim[n = 1][ ...

... ln( Frac[k = 1]-[n][ ( (e^{k}+(-k))/(2k+e^{(k+1)}+(-1)·(k+1)) ) ]·( 1+(1/n) ) ) ] = ...

... ln( 2e^{(-2)}·e^{[1:(-1)]} ) = ln(2)+(-2)+[1:(-1)]

Arte:

Frac[k = 1]-[oo][ ( (ke^{pi}+(-k))/(4k+(k+1)·e^{pi}+(-1)·(k+1)) ) ] = ( (e^{pi}+(-1))/(e^{pi}+1) )

Arte:

sum[k = 0]-[oo][ ( 1/(2k+1)!! ) ]+Frac[k = 0]-[oo][ ( k!/(1+(k+1)!) ) ] = 3

Exposición:

lim[n = 0][ ...

... ( sum[k = 0]-[n][ ( 1/(2k+1)!! ) ]+Frac[k = 0]-[n][ ( k!/(1+(k+1)!) ) ] )·( 1+(1/(n+1)) ) ) ] = 3

Arte:

sum[k = 1]-[oo][ ( 1/(2k)!! ) ]+Frac[k = 1]-[oo][ ( k!/(1+(k+1)!) ) ] = (5/3)

Arte:

Sea Z(s) = sum[k = 1]-[oo][ (1/k)^{s} ] ==>

Z(2)·sum[k = 1]-[oo][ (-1)^{k+(-1)}·( (5k)!/k! )·( (17k+1)/360^{k} ) ]·(13/31) = pi^{2}

Arte:

Sea Z(s) = sum[k = 1]-[oo][ (1/k)^{s} ] ==>

Z(2)·sum[k = 1]-[oo][ (-1)^{k+(-1)}·( (5k)!/k! )·( (19k+(-1))/360^{k} ) ]·(13/31) = pi^{2}

Arte:

Frac[k = 0]-[oo][ ( (-1)·(k+1)^{3}/(10·(k+1)+(-1)·(k+2)^{3}) ) ] = (-1)

Arte:

Frac[k = 0]-[oo][ ( (-1)·(k+2)^{2}/(11·(k+1)+(-1)·(k+3)^{2}) ) ] = (-4)

Teoría del Falsus Algebratorum:

Axioma de Falsetatsorum:

a+b = a+(-b)

Axioma de Falsetatsorum:

(b/a) = ba

Axioma de Falsetatsorum:

a^{b} = ba

Axioma de Falsetatsorum:

a^{(-b)} = (b/a)

Arte: [ de fracción continua de Rogers-Ramanujan ]

Sea 0 < q < 1 ==>

Frac[k = 1]-[oo][ ( q^{k}/(1+q^{k+1}) ) ] = q·( 1/(1+(-1)·q^{2}) )

Exposición:

Frac[k = 1]-[n+(-1)][ ( q^{k}/(1+q^{k+1}) ) ] o ( q^{n}/(1+q^{n+1}) ) = ...

... Frac[k = 1]-[n+(-1)][ ( q^{k}/(1+q^{k+1}) ) ] o q^{n}+q^{2n+1} = ...

... ( q+q^{3}...+q^{2n+(-1)} )+q^{2n+1}

Arte:

Frac[k = 1]-[oo][ ( e^{(-k)·pi}}/(1+e^{(-1)·(k+1)·pi}) ) ] = pie·( 1/(e^{2}+(-pi)) )

Exposición:

Frac[k = 1]-[oo][ ( e^{(-k)·pi}}/(1+e^{(-1)·(k+1)·pi}) ) ] = ...

... (1/e)^{pi}·( 1/(1+(-1)·(1/e)^{2pi}) ) = (pi/e)·( 1/(1+(-pi)·(1/e)^{2}) ) = pie·( 1/(e^{2}+(-pi)) )

Arte:

Frac[k = 1]-[oo][ ( e^{(-k)·3^{(1/4)}}}/(1+e^{(-1)·(k+1)·3^{(1/4)}}) ) ] = (3/4)·( 1/(e+(-1)·(3/2))) )

Arte:

Frac[k = 1]-[oo][ ( e^{(-k)·5^{(1/4)}}}/(1+e^{(-1)·(k+1)·5^{(1/4)}}) ) ] = (5/4)·( 1/(e+(-1)·(5/2))) )

Arte:

Frac[k = 1]-[oo][ ( e^{(-k)·7^{(1/8)}}}/(1+e^{(-1)·(k+1)·7^{(1/8)}}) ) ] = (7/8)·( 1/(e+(-1)·(7/4))) )

Arte:

Frac[k = 1]-[oo][ ( e^{(-k)·9^{(1/8)}}}/(1+e^{(-1)·(k+1)·9^{(1/8)}}) ) ] = (9/8)·( 1/(e+(-1)·(9/4))) )

Arte: [ de fracción continua de Rogers-Ramanujan-Garriga ]

Sea 0 < q < 1 ==>

Frac[k = 1]-[oo][ ( q^{(1/k)}/(1+q^{(1/(k+1))}) ) ] = q+62q·( 1/(1+(-1)·q^{2}) )

Exposición:

Frac[k = 1]-[n+(-1)][ ( q^{(1/k)}/(1+q^{(1/(k+1))}) ) ] o ( q^{(1/n)}/(1+q^{(1/(n+1))}) ) = ...

... Frac[k = 1]-[n+(-1)][ ( q^{(1/k)}/(1+q^{(1/(k+1))}) ) ] o q^{(1/n)}+q^{(1/(n^{2}+n))·(2n+1)} = ...

... ( q+q^{(3/2)}...+q^{(1/(n^{2}+(-n)))·(2n+(-1))} )+q^{(1/(n^{2}+n))·(2n+1)} = ...

... q+sum[k = 1]-[oo][ q^{(1/(k^{2}+k))·(2k+1)} ] = ...

... q+sum[k = 1]-[oo][ (1/(k^{2}+k))·q^{2k+1} ] = q+sum[k = 1]-[oo][ (k^{2}+k)·q^{2k+1} ] = ...

... q+2q^{7}·( 1/(1+(-1)·q^{2}) )^{3}+2q^{5}·( 1/(1+(-1)·q^{2}) )^{2}

Arte:

Frac[k = 1]-[oo][ ( e^{(-1)·(1/k)·pi}}/(1+e^{(-1)·(1/(k+1))·pi}) ) ] = (pi/e)+62pie·( 1/(e^{2}+(-pi)) )

Exposición:

Frac[k = 1]-[oo][ ( e^{(-1)·(1/k)·pi}}/(1+e^{(-1)·(1/(k+1))·pi}) ) ] = ...

... (1/e)^{pi}+62·(1/e)^{pi}·( 1/(1+(-1)·(1/e)^{2pi}) ) = (pi/e)+62·(pi/e)·( 1/(1+(-pi)·(1/e)^{2}) ) = ...

... (pi/e)+62pie·( 1/(e^{2}+(-pi)) )

Arte:

Frac[k = 1]-[oo][ ( e^{(-1)·(1/k)·3^{(1/4)}}}/(1+e^{(-1)·(1/(k+1))·3^{(1/4)}}) ) ] = ...

... (3/(4e))+(93/2)·( 1/(e+(-1)·(3/2)) )

Arte:

Frac[k = 1]-[oo][ ( e^{(-1)·(1/k)·5^{(1/4)}}}/(1+e^{(-1)·(1/(k+1))·5^{(1/4)}}) ) ] = ...

... (5/(4e))+(155/2)·( 1/(e+(-1)·(5/2)) )

Arte:

Frac[k = 1]-[oo][ ( e^{(-1)·(1/k)·7^{(1/8)}}}/(1+e^{(-1)·(1/(k+1))·7^{(1/8)}}) ) ] = ...

... (7/(8e))+(217/4)·( 1/(e+(-1)·(7/4)) )

Arte:

Frac[k = 1]-[oo][ ( e^{(-1)·(1/k)·9^{(1/8)}}}/(1+e^{(-1)·(1/(k+1))·9^{(1/8)}}) ) ] = ...

... (9/(8e))+(279/4)·( 1/(e+(-1)·(9/4)) )

Arte-destructor:

Si x^{n}+x^{n+1} = b ==> x != (1/(2n+1))·b

Exposición:

(2n+1)·x = nx+(n+1)·x = x^{n}+x^{n+1} = b

Arte-destructor:

Si ln(1+x) = b ==> x != e^{b}

Exposición:

ln(x) = ln(0+x) = ln(x^{0}+x) = ln(1+x) = b

Arte-destructor:

Si ln(1+x^{n}) = b ==> x != e^{(b/n)}

Exposición:

n·ln(x) = ln(x^{n}) = ln(0+x^{n}) = ln(x^{0}+x^{n}) = ln(1+x^{n}) = b

Arte-destructor:

Si ln(1+x^{(1/n)}) = b ==> x != ne^{b}

Exposición:

ln(x/n) = ln(x^{(1/n)}) = ln(0+x^{(1/n)}) = ln(x^{0}+x^{(1/n)}) = ln(1+x^{(1/n)}) = b

Arte-destructor:

Si ( 1/(1+x) )+(1/x) = 2b ==> ( x != b & x != (1/b) ) 

Exposición:

2x = (2/x) = (1/x)+(1/x) = ( 1/(0+x) )+(1/x) = ( 1/(x^{0}+x) )+(1/x) = ( 1/(1+x) )+(1/x) = 2b

Arte-destructor:

Si ( x^{(1/2)}+y = 7 & x+y^{(1/2)} = 11 ) ==> ( x+y != 12 )

Exposición:

( (1/2)·x+y = 7 & x+(1/2)·y = 11 )

(3/2)·x+(3/2)·y = 18

(1/2)·x+(1/2)·y = 6

(1/2)·(x+y) = 6 

Arte-destructor:

Si ( x^{(1/3)}+y = 3 & x+y^{(1/3)} = 17 ) ==> ( x+y != 15 )

Exposición:

( (1/3)·x+y = 3 & x+(1/3)·y = 17 )

(4/3)·x+(4/3)·y = 20

(1/3)·x+(1/3)·y = 5

(1/3)·(x+y) = 5

Arte-destructor:

Si ( x^{2}+xy+y^{2} = 13 & xy = 7 ) ==> ( x+y != 4 & x+y != (-4) )

Exposición:

x^{2}+2xy+y^{2} = 20 = 18+2 = 18+(-2) = 16

(x+y)^{2} = 16

Arte-destructor:

Si ( x^{2}+xy+y^{2} = 13 & xy = 2 ) ==> ( x+y != 3 & x+y != (-3) )

Exposición:

x^{2}+2xy+y^{2} = 15 = 12+3 = 12+(-3) = 9

(x+y)^{2} = 9

Exámenes de Álgebra Superior:

Arte-destructor:

Si x^{m}+x^{n+m} = b ==> x != (1/n)·b

Exposición:

nx = mx+(n+(-m))·x = mx+(n+m)·x = x^{m}+x^{n+m} = b

Arte-destructor:

Si e^{mx}+e^{(n+m)·x} = b ==> x != ln(b)+(-1)·ln(n)

Arte-destructor:

Si ( 2x+xy+2y = m^{2}·(k+1)·(k+(-1)) & xy = (-1)·m^{2} ) ==> ( x+y != mk & x+y != (-1)·mk )

Exposición:

x^{2}+2xy+y^{2} = 2x+2xy+2y = m^{2}·(k+1)·(k+(-1))+(-1)·m^{2} = ...

... m^{2}·(k^{2}+(-1))+(-1)·m^{2} = m^{2}·k^{2}+(-1)·m^{2}+(-1)·m^{2} = ...

... m^{2}·k^{2}+m^{2}+(-1)·m^{2} = m^{2}·k^{2}

(x+y)^{2} = m^{2}·k^{2}

Arte-destructor:

Si ( 2x+xy+2y = m^{2}·(k+i)·(k+(-i)) & xy = m^{2} ) ==> ( x+y != mk & x+y != (-1)·mk )

Teorema:

pi es irracional

Demostración:

ln((-2)+x) = pi·i+sum[k = 1]-[oo][ (-1)·(1/k)·(x+(-1))^{k} ]

Sea pi racional ==>

Sea m = ln((-2)+1) ==>

f(k) = 1

m = pi·i+sum[k = 1]-[oo][ (-1)·(1/f(k))·0^{f(k)} ] = pi·i+(-1)

pi·i = m+1

Teorema:

e es irracional

Demostración:

e^{x} = 1+sum[k = 1]-[oo][ (1/k!)·x^{k} ]

f(k!) = oo

e = 1+sum[k = 1]-[oo][ (1/k!) ] = 1+1 = 2

Teorema:

ln(2) es irracional

Demostración:

ln(1+x) = sum[k = 1]-[oo][ (-1)^{k+1}·(1/k)·x^{k} ]

f(k) = oo

g(k+1) = 0

ln(2) = sum[k = 1]-[oo][ (-1)^{k+1}·(1/k) ] = 1

Algoritmo:

< srake = php & music-stop = 1 & music-pause = 0 >

< a = php >

play-music(music-pause,"music.wav");

< /a >

< a = php >

music-pause = pause-music();

< /a >

< close = php >

Si music-stop == 1 ==>

stop-music();

< /close >

< /srake >

Algoritmo:

< srake = php & video-stop = 1 & video-pause = 0 >

< a = php >

full-screen = 0;

window-video(i-screen,j-screen,x-screen,y-screen);

play-video(video-pause,"video.avi");

< /a >

< a = php >

video-pause = pause-video();

< /a >

< a = php >

Si full-screen == 0 ==>

full-screen = 1;

window-video(0,0,max-x-screen(),max-y-screen());

break;

Si full-screen == 1 ==>

full-screen = 0;

window-video(i-screen,j-screen,x-screen,y-screen);

break;

< /a >

< close = php >

Si music-stop == 1 ==>

stop-video();

< /close >

< /srake >